If you'd like a more specific time period, I'm particularly interested in very early city states like Uruk. My general impression was that cavalry was rare in ancient Mesopotamia (and by extension so was mounted travel), but I recently saw an Assyrian relief at the British Museum that depicted a mounted warrior.
Mounted travel was pretty much nonexistent for most of Mesopotamian history, but cavalry did become prevalent during the rise of the Neo-Assyrian empire. It was a major factor in their military success after its introduction. Prior to that horses were only known through sparse contact with nomadic groups from the Northeast. I don't know of any animals that were ridden before horses, but onagers were commonly used beasts of burden along with camels and a few other miscellaneous creatures.