I did find an excellent adaptation of Gilgamesh, by Bernarda Bryson. Although it's quite suitable for a read-aloud to children, it's not at all childish. (A series of Gilgamesh picture books I got was too childish even for children.) DOB read it and declared that Gilgamesh is henceforth his favorite mythological character. His struggle is not just against monsters but against his own limits. The illustrations are cleverly done, too–they're closely based on suitable artwork from the Near East.
I really haven't found many good children's books on Babylon or Assyria. But that may be a good thing; after all, one or two good books are better than a whole plethora of mediocre ones. Reading two or three of the tales of ancient Babylon, plus a few chapters from a history book and a few factual books on the life and times might be more memorable than the overload temptation that will face us with Egypt, Greece and Rome.
I've been debating whether I want to include a study of the Americas or India and China in this first year. After reading a book on the Mound Builders and other North American civilizations, though, I'm convinced that I do. After all, the Mound Builders operated right in our back yard. Taking some time out in late September (when the weather is likely to be perfect) to visit the local sites may be a great way to get a tangible connection with the archeological methods and findings of ancient civilizations far away.