It's a remarkable conjecture, but not one to take seriously. (I can think of at least half a dozen reasons for extreme skepticism off the top of my head.) Still, it has obvious possibilities for science fiction. Return to bicamerality is, for instance, a plot device in two (unrelated) novels by Neal Stephenson. [Note with spoilers for those books.] I'm sure there are stories with bicameral aliens (much more plausible neurologically), but I've not run across any. Turtledove's novel is the first work of fiction I've encountered which incorporates an obvious improvement upon Jaynes's original hypothesis.
The ancients, after all, don't just show the gods telling people ``do this'' or ``boy, what we have here is a failure to communicate''; they show the gods taking actual physical actions. (Apollo doesn't just tell people what to do, he shoots them, and other gods.) If we accept one set of reports on Homer's say-so, why not both? This means that the gods were actual physical beings, looking more or less like human beings but much more powerful. (It also means that ghosts were around, hungry for offerings of blood and wine.) Instead of very implausible changes to the workings and anatomy of the human nervous system, the breakdown of ``bicamerality'' would mean that people somehow or other got out from under the gods' thumbs.
There are two ways of taking this simplification, at least for purposes of fiction. In science fiction, one would go the ``chariots of the gods'' route, and assume that the deities were just creatures with technology sufficiently advanced that it looked like magic to clod-hoppers in the Fertile Crescent and the Balkans. In fantasy, on the other hand, magic is magic, and supernatural beings are supernatural beings. Turtledove has gone for fantasy.
Between the Rivers is set fairly early in the escape of mortals from divine domination --- which Turtledove starts long before Jaynes does. (Just when is ambiguous; there are, thankfully, no captions reading ``Sumeria, 777 B.C.'') Each of the cities in the land between the rivers has its own god or goddess, who mentally dominates its inhabitants --- ranging from reading their minds and issuing irresistible commands, through looking out through their senses, to flat-out possession. The one exception is the city of Gibil, where, in exchange for rich offerings and more than usually fulsome flattery, the god, Engibil, is content to let a human king rule, and leave the minds of the citizenry alone. Consequently, the Giblut show more smarts and initiative than the rest of the people in the known world, and are much better at three recent activities which have no god in charge of them --- metal-working, writing, and trading. Our hero is a young merchant from Gibil, setting off on a caravan to Alashkurru, the mountains to the north of the land between the rivers, where he plans to trade bronze weapons and dyed clothes for raw copper ore. But the other gods are not very happy to see their slaves getting ideas from the Giblut...
Suffice it to say that the plot is a (reasonably straightforward) hunt for a McGuffin, with a war between Gibil and a neighboring city tossed in. Details of daily life are pretty faithful to what the archaeologists say about ancient Sumer (Turtledove's a historian in mundane life, after all), and are conveyed with a minimum of info-dumping. Proper names are all (I think) made up, but sound right --- those in Alashkurru sound Hittite or Hurrian --- and the dialogue (but not the narrative) is modeled on that in authentic cuneiform texts. (This is almost certainly not how actual Sumerians spoke to each other, of course --- a point Jaynes ignores.) The characters are rather flat; this might be defended as a world-building feature, since they've only had a few generations, at most, of mental autonomy, but still doesn't do much for the novel. Nor is this excuse available for the gods, not one of whom is sharper than a bag of hammers.
On balance, it was an enjoyable way to pass a sunny weekend afternoon, and good enough that I'll probably read more Turtledove in the future, but not so good that I'll press it upon all my acquaintance, or rush out to get the rest of his books.
The cover-artist mistook this book for a Biblical movie epic in which the Levant is inhabited by Danes; otherwise, the production is excellent, and typo-free.