The answer is: no, we don't know definitively. There are several theories currently, none of which has been conclusively proven and all of which have some merit:
that the peoples who eventually evolved into the "Sumerians" of the 4th millennium came from Africa; DNA studies have as yet been inconclusive on that, though there is some good DNA-based evidence that Africans were in the fertile crescent in some form or another in prehistory.
linguistic evidence of place-names and geographical features in Mesopotamia suggest an origin in West Asia, but not from the same group(s) that later produced the Semitic and Indo-European language families. This isn't my area of expertise, but I believe it is generally accepted these days that Sumerian was a language isolate which interacted with and influenced Akkadian.
there is some limited DNA-based evidence to suggest that the "proto-Sumerians" came from the Indus Valley Civilization, related to the Dravidians (I'm not a huge fan of this one, myself; I would expect much more typological continuity in some form or another, be it pottery technology, artistic motif—such as there is in this period—, agricultural profile, anything really; a couple of random bones with DNA markers does not excite me like it does some others)
another puts them in association with the Hurrians, and thus from the Caucuses in origin (evidence is pretty thin, in my opinion)
yet another connects them with the Arabian bifacial culture, named for hand axe technology. They are purported to have lived on portions of the Arabian coast of the Persian gulf which became submerged at the end of the Ice Age (I don't know anything about this one, myself)
last but certainly not least, the simplest and most logical explanation, in my mind: that they derived from the "proto-Euphratean", which in turn derived from the Samarra culture of northern Mesopotamia; they were the first organized society in Mesopotamia and, in their Chalcolithic heyday, demonstrated many of the basic facets of later Neolithic civilizations: social hierarchical organization, long-distance trade, large-scale agriculture and agricultural geography (like draining swamps, constructing terraces, and clearing flora), significant developments in several different industries (metallurgy, leather-working, pottery, textiles, etc).
For this last, see Charles Maisels, The Near East: archaeology in the 'cradle of civilization (Routledge 1993), especially chapters 6, 7, and 9.