Sumerian was spoken in South Babylonia, which is the part of Mesopotamia1 situated between the lower courses of the rivers Euphrates and Tigris and including adjacent areas across the rivers (what is now southern Iraq). In the east, the language area reaches up to the Zagros mountains, in the west up to the Syrian desert. The coordinates are approximately: 40° – 50° longitude, 30° – 40° northern latitude.
http://encarta.msn.com/media_461553469_761576369_-1_1/Kingdom_of_Sumer.html
A time-line with historical maps is on the website of ECAI Iraq.
Sumerian texts have also been found in Turkey, Egypt, Syria and Iran.
Up to -3.500, the climate was much more humid, so that the rivers covered more space, and the delta (or the Persian gulf) reached up to Ur in the north. Subsequent drier, mediterranean climate generated fertile alluvium, rendering a denser population possible (Strommenger 1979:8).
1 Greek mesopotamia ‘land amidst the rivers’