I went for Persia, mostly the same reasons as lanzelot. It is a good all-around civ with a solid trait combo and an excellent UU. You can compete on any level for any victory condition. Probably one of my favorite civs as well as one of the best.
The Babylonians are good for culture games, with tons of half price culture buildings. Average to bad UU (offensive unit with defense bonus).
The Greeks are good at science and at peaceful building in the early game, Hoplites will keep you safe.
The Egyptians can be a fair civ, they don't really stand out much. Somewhat map dependent UU, although good value for the cost.
Romans are the first civ I played, but are pretty average overall. Odd trait combination and UU that may not need extra defense.