In The World of Odysseus by Moses Finley and he throws out a line in chapter one:

Homer was so far from Socrates that he was not even cognizant of man as an integrated psychic whole.

The context is:

One measure of man's advance from his most primitive beginnings to something we call civilization is the way in which he controls his myths, his ability to distinguish between the areas of behavior, the extent to which he can bring more and more of his activity under the rule of reason. In that advance the Greeks have been pre-eminent. Perhaps their greatest achievement lay in their discovery-more precisely, in Socrates' discovery---that man is "that being who, when asked a rational question, can give a rational answer." Homer was so far from Socrates that he was not even cognizant of man as an integrated psychic whole.

I looked up that phrase to see what exactly he might be talking about. Almost all of the results concern the Greeks or Romans.

From Homer in Performance: Rhapsodes, Narrators, and Characters:

The critical bibliography on character monologue in Homeric epic is extensive. Scholars have been divided as to whether to see merely a convention or dramatic technique for representing a character's inner thoughts, or to take the talking thumos as a separate entity, an alter ego that represents a not-yet-integrated psychic whole. Other scholars see the Homeric monologues as evidence of Homeric psychology in general and use them to study Homeric decision making as it prefigures later Aristotelian and Stoic theories about human rationality and motivation.

It sounds close to what Jaynes was saying. The Greeks of Homer were split between rational and irrational selves.
Jaynes would have denied that Romans were bicameral or were not integrated psychic wholes. But the phrase is used in The Roman Gaze: Vision, Power, and the Body:

To this end, the Roman incorporated others into himself or herself as witnesses and ideals. "I tell my son to look into the lives of all others as if into a mirror and to take from others a model for himself."

Roman honor, then, was a way of self-regarding as well as other-regarding. Honor required self-splitting; one needed to be, at all times, both the watched and the watcher. For the Roman, there could be, finally, no integrated psychic whole, no stable notion of self. If a Roman had a sense of "integrity" it was one built, paradoxically, on the dividing of the self. Cicero speaks of the self-control needed to resist shameful reactions to pain: "I'm not exactly sure how to say it, but it is as if we were two people: one who commanded and one who obeyed.”

And Roman Honor: The Fire in the Bones:

The provisional and contested nature of reality (including the reality of one's being) and the immediacy and particularity of experience infused all Roman ways of thinking. The Romans did not have an "integrated psychic whole," and they tended not to synthesize or carefully correlate parts to a whole. Boundaries and obligations tended to accumulate and to overlap without being codified or systematized. The Romans were slow to deduce principles or create Utopias. There is a reason that modern philosophers and political theorists ignore the Romans: though rich and complex, the thought of the Romans is not easily translated into the categories or linearities of modern Western thought, with its rigid dichotomies and principle of noncontradiction.

Cicero comes right out and admits his bicamerality, but it is clearly not the same type that Jaynes wrote about. It is an artificial or voluntary version, though the second quote suggests they were different in more ways. They seemed to recognize an “animal” part of man that reacted to a stimulus and a “rational” part that was able to modulate that reaction. Jaynes says the Greeks and other ancient people understood that rational part to be gods or kings or ancestors while other authors say the Romans intentionally personified that part as a respected member of society, just to give it a little more force. But they were still split in a way we are not. They maybe had a better understanding of themselves, were able to decouple their actions and reactions from their thinking selves and analyze them. Nowadays our rational and animal parts are a jumble. People come to identify with their reactions and think any criticism of it is an attack on their self. I think that is a big cause of depression and other mental disorders. People don’t know why they react the way they do and feel out of control. They go to therapy to replace what would have been a hallucinated god three thousand years ago. The therapist walks them through their feelings because we forgot how to do that ourselves. That is how I read it anyway. It isn’t too important, the point is that these people believe the ancients had fundamentally different psyches than modern man. I don't actually know if even modern man has an integrated psychic whole.