What is the meaning and implication of Lennox's speech in act 3, scene 6 of Macbeth?

Well done for spotting the gap between appearance and reality in this speech, and the sarcasm of Lennox as he comments upon recent occurrences in the kingdom of Scotland! As Lennox is seen talking to a Lord, he discusses his suspicions of what has happened and the true nature of events. However, although he reports the known facts, if we look carefully and what he says and how he says it, we can see that he is being incredibly ironic, and he does not actually mean what he says at all. Note his comment on Macbeth and his actions in killing Duncan's two henchmen:

So that, I say,
He has borne all things well, and I do think,
that, had he Duncan's sons under his key
(As, and't please Heaven, he shall not), they should find
What 'twere to kill a father; so should Fleance.

Clearly, Lennox does not believe Macbeth has done "all things well," as he believes that if Macbeth had access to Duncan's sons and to Fleance that they would soon meet the fate of others that Macbeth has had close contact with. Throughout this entire speech, Lennox is discussing his suspcions of Macbeth, ironically underlining the difference between Macbeth's actions and his real motives.

In Macbeth, Act 3, Scene 6, why is Lennox cautious in his speech and does he criticize Macbeth?

When Lennox first appears in Act 2, Scene 3, he is obviously a very young, naive lad who is ill at ease and unsure of himself. When Macduff goes into Duncan's chamber leaving Lennox alone with the grim and silent Macbeth, the young man feels obliged to talk about something, so he talks about the weather.

The night has been unruly.Where we lay,
Our chimneys were blown down, and, as they say,
Lamentings heard i’ the air, strange screams of death,
And prophesying with accents terrible
Of dire combustion and confused events
New hatch'd to the woeful time. The obscure bird
Clamor'd the livelong night.Some say the earth
Was feverous and did shake.

Then Shakespeare uses this character again in Act 3, Scene 6, to convey an impression of how much has happened to Scotland since Macbeth's coronation. There has been a striking change in young Lennox which can only be attributed to the traumatic events he has experienced and witnessed in a few short years. Now he seems grown up, sophisticated, worldly wise. He has learned to be guarded in his speech and not to babble everything he knows. Macbeth has secret agents everywhere. In Act 3, Scene 3, Macbeth tells his wife:

There's not a one of them but in his house
I keep a servant fee'd.

Lennox has learned innuendo, imputation, and discretion. His entire speech to this unnamed Lord is in strong contrast to the way he thought and spoke in Act 2, Scene 3.

Shakespeare's purpose is to show how much things have changed in Scotland since Macbeth took the throne. In Act 4, Scene 3, Ross tells Malcolm and Macduff about conditions at home.

Alas, poor country,
Almost afraid to know itself! It cannot
Be call'd our mother, but our grave. Where nothing,
But who knows nothing, is once seen to smile;
Where sighs and groans and shrieks that rend the air,
Are made, not mark'd; where violent sorrow seems
A modern ecstasy. The dead man's knell
Is there scarce ask'd for who, and good men's lives
Expire before the flowers in their caps,
Dying or ere they sicken.

Everybody now feels sure it was Macbeth who murdered Duncan and intended to murder Malcolm and Donalbain. Everybody knows it was Macbeth who had Banquo murdered and tried to have Fleance murdered at the same time. Everybody hates Macbeth for his criminal acts--and this hatred, bordering on outright rebellion, has forced Macbeth to become a tyrant who rules by fear. All of this is shown through the formerly callow young Lennox, who not only imparts a great deal of information but represents the bitterness and disillusionment of all the thanes and most of the Scottish people. This is Shakespeare's way of encapsulating time, atrocities, dissatisfaction, rumors, and countless incidental events into one single character's monologue.

In Shakespeare's Macbeth, Act 3.6, Lennox may be being politically careful in his speech.  He says nothing that could cause him to end up like the murdered people he mentions--Duncan and Banquo--or the person he suggests is now in danger--Macduff--due to his absense from the "tyrant's" feast. 

 The key to understanding Lennox's true feelings in the speech is "tyrant."  Lennox considers Macbeth a tyrant.  It is hard to hear that and then still conclude that Lennox really believes Malcolm and Donaldbain killed their father, and, even more unlikely, that Fleance killed Banquo.

It is much more likely that Lennox is using sarcasm, if the speech is performed with a sarcastic tone of voice, or at least irony, if the tone of voice is not sarcastic.

Shakespeare often wrote ambiguously, leaving the words of a character to be interpreted by directors and actors.  This is probably one of those instances.  One thing is relatively certain, though:  Lennox is questioning the theory that the deaths he mentions were the result of sons killing their fathers. 

Why is Lennox cautious in his speech in Act 3, Scene 6 of Macbeth, and does he criticize Macbeth?

Lennox IS quite careful in his speech. In Act 3, scene 5, Macbeth tells his wife that he has spies in every lord's house:

There's not a one of them but in his house
I keep a servant fee'd.

Consequently, it pays to be cautious in criticizing the king, especially when the consequences can be severe.  Macbeth has turned against Macduff not only because of the apparition's prophesy, but also because his spies have discovered Macduff's disloyalty.  In this speech, Lennox is ironically critical of Macbeth.

His words taken at face value can go for praise. He says in reference to Macbeth's killing Duncan's guards:

Did he [Macbeth] not straight,
In pious rage, the two delinquents tear,
That were the slaves of drink and thralls of sleep?
Was that not nobly done?  Ay, and wisely too;
For 'twould have angered any heart alive
To hear the men deny't.

A spy reporting Lennox's words might have a difficult time proving that Lennox is being critical at all.  Yet, Lennox's irony is far from subtle.  Within his speech, Lennox implies that it is ludicrous to believe that Duncan's sons killed their father.  He suggests in a darkly humorous way that Macbeth will probably try to pin the murder of Banquo on Fleance, because he, like Donalbain and Malcolm, fled.  Lennox predicts that Macbeth would kill Donalbain, Malcolm, and Fleance if they were in his custody.  He implies that killing Duncan's guards was a wise act for Macbeth because they would have denied taking any part in the killing of Duncan--why?  because they were innocent!

At the end of this scene, though, Lennox drops his irony and sarcasm.  He more directly refers to Macbeth as a "hand accursed" that is ruling "our suffering country."  He wants Macduff to exercise caution and prays that Macduff will be able to get help from England to overthrow Macbeth.

What does Lennox imply about Macbeth's actions in Act 3, Scene 6, in everyday English?

Lennox is an interesting character who has a small role but one that represents the noble's side of the situation at Macbeth's castle as witness Macbeth's actions and reactions. Earlier, in Act II.iii, Lennox has the role of describing the strange atmosphere during the night of Macbeth's foul murder plot. Lennox speaks of the night as being "unruly" with "dire combustion and confused events." This positions Lennox as the observer and reasonable commentator on unclear and unexplained events. He also helps Macbeth tell the horrible news of Duncan's murder to Malcolm

MALCOLM
    O, by whom?

LENNOX
    Those of his chamber, as it seem'd, had done 't:
    Their hands and faces were an badged with blood;
    So were their daggers, which unwiped we found
    Upon their pillows: (II.iii)

In III.iv, at the banquet, Lennox assists and has compassion for Macbeth's disconcerted state, showing him seat, saying it is "reserved" for him and asking what troubles Macbeth: "What is't that moves your highness?" This establishes Lennox in the role of unbiased, objective questioner: he is puzzled by what he sees (especially since what he does not see is Banquo's ghost) and objectively notes Macbeth's strange behavior and reasonably inquires about what is troubling Macbeth (Lady Macbeth does the same thing but not quite so reasonably).

I III.vi, we encounter Lennox part way through a conversation with another Lord. Lennox doesn't say anything accusatory or intentionally hint at anything accusatory about Macbeth. What Lennox does do is rehearse events out loud, with the other Lord, in the fashion in which he is turning the thoughts round in his own mind. After hearing what Malcolm and Macduff are doing in the court of King Edward of England (raising an army to go up against Macbeth), he sees things in a clearer light and switches to the analysis of events that condemn Macbeth's behavior as he prays that some angel might be sent to aid Malcolm and Macduff and that order might be restored to the land:

LENNOX
    ... Some holy angel
    Fly to the court of England and unfold
    His message ere he come, that a swift blessing
    May soon return to this our suffering country
    Under a hand accursed!

PARTIAL CONTEMPORARY ENGLISH PARAPHRASE OF LENNOX III.vi

• Our thoughts agree. Macbeth has borne with events is a strange manner [III.iv]. Macbeth was grieved about Duncan's death and Banquo's death was seemingly his own fault for being out walking too late ... Macbeth was so outraged .... Didn't this show Macbeth to be noble and wise too? Anyone feels the same outrage that he acted upon.

• Thus it is that I say Macbeth has taken all these things well and behaved in an expected manner. ... But now! I hear Macduff is out of favor with Macbeth because he spoke out against Macbeth and decline to feast with Macbeth. Can you tell me where Macduff is? ...

• Well, ... Macduff might well be cautious and keep his distance. If an army truly is needed ... may a swift angel make it all clear to England and may a swift blessing of retaliation against Macbeth be brought to our land to restore peace.

In Macbeth, why is Lennox cautious in his speech and does he criticize Macbeth?

Lennox was introduced in Act II, Scene 3, when he came with Macduff to wake King Duncan and was standing outside the King's chamber trying to make conversation with Macbeth when Macduff discovered the murdered body. At that time Lennox's dialogue characterized him as very young and naive. He was obviously starting out in his career as a courtier and soldier. Then he dropped out of sight until Act III, Scene 6. Shakespeare uses this character very adroitly to show that a lot of time has passed and that there have been a lot of changes in the country under Macbeth's rule. Lennox has become much older and wiser. He has learned to talk ironically, often saying the exact opposite of what he means. The effect is to show the audience that Macbeth is a villain and has become a tyrant. Macbeth rules by fear, not unlike Hitler in Nazi Germany, or Stalin in Soviet Russia, or Robespierre during the French Revolution. Macbeth has spies everywhere, even acting as servants in private households. One might estimate that something like eight or ten years have passed since Macbeth murdered Duncan and was crowned king. The whole country lives in fear. Everyone has to watch what he says, because if he says the wrong thing and it gets back to Macbeth, he could disappear and never be heard from again.

In Act IV, Scene 3, Ross, who is newly arrived in England, will describe conditions in Scotland in more explicit language to both Macduff and Malcolm: 

Alas, poor country,
Almost afraid to know itself! It cannot
Be call'd our mother, but our grave. Where nothing,
But who knows nothing, is once seen to smile;
Where sighs and groans and shrieks that rend the air,
Are made, not mark'd; where violent sorrow seems
A modern ecstasy. The dead man's knell
Is there scarce ask'd for who, and good men's lives
Expire before the flowers in their caps,
Dying or ere they sicken.

Lennox is only saying, in closely guarded language, what everyone now knows. Macbeth murdered Duncan. He had Banquo murdered by agents and tried to have Fleance murdered along with his father. Macbeth would love to get his hands on Malcolm, Donalbain, and Fleance so that he could have them executed. He would accuse Malcolm and Donalbain of bribing Duncan's attendants to kill him in his sleep, and he is quite capable of accusing Fleance of being responsible for his father Banquo's murder.

Shakespeare frequently uses scenes like III.6 to convey information to his audience. The most important news comes from the unidentified Lord when Lennox concludes his monologue and asks:

I hear Macduff lives in disgrace. Sir, can you tell
Where he bestows himself?

The Lord informs Lennox, and the audience, that Macduff is living at the English court with Duncan's son Malcolm, who fled there after his father's assassination. Both are trying to persuade King Edward to help them overthrow the tyrant Macbeth and install Malcolm as King of Scotland. This foreshadows the climax, in which an army of 10,000 English soldiers and Scottish defectors with Malcolm and Macduff in their vanguard invade Macbeth's realm and Macduff kills Macbeth in hand-to-hand combat.