Their language is now understood to have many Dravidian elements, although it was advanced well beyond all such languages at the time writing was invented ~ probably driven by writing itself. At the same time the Sa'ami languages are also quite arguably Sumerian cognates ~ but the problem is that it's only been within the last few hundred years any of them were rendered into writing, while at the same time they've been heavily influenced by Indo-European and Fenno-Ughric languages.
Archaeologists in Iran are working on an ancient pre-Sumerian settlement/city/town/whatever that appears to connect ancient Mesopotamia with ancient South India.
Most of the above was predicted shortly after the first translations were made of early Sumerian texts ~ and simply on the basis that the Sumerians discussed Ice Flows, Glaciers and sunny beaches with palm trees (at least the way the early Hungarian translators understood the texts).
The Great Depression and WWII/Communist occupation/etc. left careful examination of Sumerian materials in the lurch for a very long time. Scholars are just now getting back into the wealth of untranslated materials.
Cuneiform decipherment:
http://en.wikipedia.org/wiki/Cuneiform_script#Decipherment
http://en.wikipedia.org/wiki/Sumerian_language