When flowing Cups run swiftly round    
   With no allaying Thames,
Our careless heads with Roses bound,
   Our hearts with Loyal Flames;
When thirsty grief in Wine we steep,
   When Healths and draughts go free,
Fishes that tipple in the Deep
   Know no such Liberty.

Stanza 2 is a little difficult to understand because of the obscure language in lines 1 and 2. Some understand "flowing Cups" that "run swiftly round" as a reference to fish in the Thames river. This is because of (1) the line 2 reference to the Thames and (2) the lines 7 and 8 reference to fish. Yet, none of the other 3 stanzas use allusions that are so obscure; others are easier to recognize, e.g., "unconfinèd wings" of love; "like committed linnets";  "Walls do not a Prison make." Therefore it makes sense that "flowing Cups" is an equally recognizable allusion (i.e., reference to a known concept).

A more logical way to understand the phrase is in light of the standard metonymy in which "flowing Cups" stands for energetic wine drinking. This idea is supported by the line 5 reference to "Wine" and by the subsequent allusions to toasts ("draughts") and to non-alcohol-drinking fish ("tipple"). The line 2 allusion to water, "Thames," is a complicated allusion as the Thames is called the "no allaying Thames." To "allay" means to reduce the intensity of; to diminish or calm. Thus, in reference to wine cups in which wine continually flows, "With no allaying Thames" means the flowing wine is not deintensified by being mixed with water.

When flowing Cups run swiftly round    
   With no allaying Thames,

Lines 3 and 4 describe the state of the drinkers included in "our." They have their heads joyously "bound" in celebratory roses; they have their hearts ignited in loyalty by "Loyal Flames." Though not stated, it may be inferred that they are drinking toasts of celebration and love to King Charles I, to whom their hearts are ever "Loyal" and the grief for whom they have momentarily set aside.  

Our careless heads with Roses bound,
   Our hearts with Loyal Flames;

Lines 5 and 6 carry this further and describe how they disregard "griefs" and change grief for freely given toasts to each others' health.  Lines 7 and 8 provide the comparison, in the form of an analogy, that describes how much freedom, or "liberty," this celebration provides the partakers. Lovelace says that not even "fishes" who swim without restraint in the sea know the kind of liberty these have who engage together where "flowing Cups run swiftly round" and who have "careless heads with Roses bound"; (in this usage, "careless" means without care, worry, or "grief.")

Fishes that tipple in the Deep
   Know no such Liberty.