Okay, I couldn't figure this out (it was turning into a chicken-or-the-egg scenario in my mind), so I looked in one of my old textbooks, Understanding Early Civilizations by Bruce Trigger. In this book, he looks at the development of "ancient Egypt and Mesopotamia, Shang China, the Aztecs and adjacent peoples in the Valley of Mexico, the Classic Maya, the Inka, and the Yoruba." Of that list, "Egypt and Mesopotamia were the only early civilizations. . .that supplemented human agricultural labour with that of domestic animals. Oxen- and donkey-drawn ploughs were present from an early period. Draft animals are estimated to have resulted in a 50% reduction in the human labour needed to grow grain, and this permitted small groups of men to work large, monocropped fields." (281) The wording of that sentence kind of indicates that grain-producing was going on without the help of animals, and that the use of beasts of burden just made it more efficient and widescale.

As for the other civilizations, they were farming with a variety of tools, be they wood, shell, various metals, etc. He mentions handploughs and other things like hoes and such. So I guess it depends on what kinds of grain you mean, since each of these civilizations grew different crops. I'd say at least some grains were being produced without the help of animals, although I'm sure others were developed post-domestication of draft animals.

If you'd like to read the whole chapter, I found the book on Google Books: http://books.google.com/books/about/Understanding_early_civilizations.html?id=ZEX-yZOAG9IC

Just hit the preview button. The chapter is 14: Food Production, which begins on page 279 :-)