I think that the largest thematic idea with which Achebe seems to be the most preoccupied is the notion of change in the modern setting. Achebe places Okonkwo in a condition where a strong sense of confusion about what the traditional notions of the good flies in the face of modern complexity. From this particular vantage point, Achebe is thematically able to pivot to the larger idea of how to appropriate the modern sense of understanding in a world of total change. When Okonkwo returns to his village, all has changed, with everything that once defined meaning having "fallen apart." It is here where I feel there is the greatest thematic preoccupation in terms of how individuals are able to appropriate the idea of change in the modern condition to their own lives. For Okonkwo, this notion of change is something that impacts him on a cultural and personal level. It is this particular idea that becomes the central force for both the protagonist, and also for the author. This preoccupation is driving the narrative and something that makes the work incredibly relevant.
Thematically, Things Fall Apart is concerned often with the idea that morality has various frameworks. Another way of putting this idea would be to say that what is morally right is determined by context (and not by a set of apriori absolutes).
We can see this thematic notion in several iterations relating to Okonkwo particularly, to the Igbo people more generally and to the Western colonialists at the end of the novel.
Nowhere is this idea dealt with more directly than in the scene following Okonkwo's death. While burial is necessary, Okonkwo's own tribesmen cannot bury his body. The code of of conduct for the tribe (essentially a moral code) will not allow it. They are forced to ask the Commissioner and his men to bury the body.
"'Why can't you take him down yourselves?' he asked.
'It is against our custom,' said one of the men. [...] His body is evil, and only strangers may touch it. That is why we ask your people to bring him down, because you are strangers.'"
The moral code that prohibits the tribe from touching Okonkwo's body allows others to touch the body. Morality, clearly, is determined according to specific contexts. What is wrong for some is acceptable for others.
This theme is not merely a facile or superficial comment on the rigors of tribal customs. As the theme is extended as an investigation of the ways that differing cultures often fail to understand the specific morality of other cultures, presuming that the moral codes of the local culture is actually absolute and should prevail to govern the behavior of all groups. This is the attitude of the colonists, underpinning their entire enterprise of spreading Christian doctrine and Western ways of life.
In other instances in the novel, we see Okonkwo's behavior in light of the moral codes of his tribe and, as readers, are forced to reconcile the idea that certain actions in the Igbo culture are moral/right despite any prejudice we might bring to the book to the contrary. Even within the Igbo culture, context functions as a significant factor in determining what is morally right and what is morally wrong.
Although [Okonkwo] suffers severe remorse over his complicity in the first killing [of Ikemefuna], the second one has been an accident. Yet the tribe punishes Okonkwo with banishment for the second killing and not at all for the first.
The idea that morality is to be judged from within a culturally defined framework of values (and, perhaps, history too) is at the heart of the commentary made in Achebe's novel. This is not a statement of moral relativism in the sense that all actions might be justified given the right contextual "spin." Rather, the statement expressed in and through the novel's concern with the theme of moral frameworks is related to a post-colonial challenge to presumptions of superiority made by dominant cultural forces.
The proverb, "might does not make right," resonates with this thematic thread in Things Fall Apart as "right" is determined by means of specific contexts, specific local mores and by particular cultural assumptions. In attempting to subdue the moral perspective of the Igbo people, the colonists represent a lack of humility and a failure to understand the various sources of morality and moral thought.