Egyptians:
controlled irrigation (I'm not sure if they invented this but it would definitely have made their civilization possible)
writing system
Babylonians:
the Code of Hammurabi (the foundation of the law as independent from the will of the ruler)
literature (the Babylonians had a large body of literary works including law, science, mythology, and correspondence)
Phoenicians:
The alphabet (a phonetic way of writing)
The bireme, a sailing vessel which allowed them to travel and trade throughout the Mediterranean. They possibly went as far as Great Britain, may have circumnavigated Africa, and sailed down the Red Sea
Greeks:
Democracy and drama are the two traditional answers to this question. Philosophy (knowledge for its own sake) and public education were important innovations too.
Romans:
The organization of the Legion, which included light infantry, cavalry, and the heavy infantry
I think they invented concrete; certainly they used it a lot in building those celebrated Roman roads!
Aqueducts
Allowing the conquered people in the empire to retain their distinct cultures but still allowing them to become full citizens
Israel:
I was going to say "monotheism," but that concept came from the Egyptians. I'm pretty much drawing a blank on this one.
This was a fun question!
