Wyse and Winkleman:
"During the 7th millennium farming villages, hitherto confided to the Zagros mountains, began to appear in the rain-fed north Mesopotamian plain. Within a few centuries the development of irrigation allowed settlement to spread into central Mesopotamia, ultimately reaching the rich alluvial lands of the south, where the first cities were to eventually emerge." (p. 98. "Mesopotamia: Towards Civilisation." Elizabeth Wyse & Barry Winkleman, et. al. Past Worlds, Atlas of Archaeology. Ann Arbor, Michigan. Borders Press in association with HarperCollins.  1999. ISBN 0-7230-1054-4. paperback)
Flaherty on the below map:
"This map of the ancient Near East locates some of the prehistoric sites on the northern and the eastern borders of Mesopotamia described in the chapter. Here, in villages in the hills, lived the early farming peoples who may later have migrated into the Mesopotamian plain, where such settlements as Uruk, Ur, and Eridu evolved into the world's first cities." (p. 57. "Milestones On The Road To Civilization." [Chapter Two]. Thomas H. Flaherty. Editor.
Sumer: Cities of Eden. Alexandria, Virginia. Time-Life Books. 1993. ISBN 0-8094-9887-1)
The late Professor Kramer noted that the first inhabitants of what was to later be called Sumer, were not Semitic Sumerians. A study of the language preserved on the cuneiform texts suggested a non-Semitic peoples had laid the foundations of civilization in Lower Mesopotamia. Archaeology would trace these peoples to the hill country in northern Mesopotamia (cf. the above map):
"Sumer, or rather the land which came to be known as Sumer during the third millennium BC, was probably first settled sometime between 4500 and 4000 BC -at least this was the consensus of Near Eastern archaeologists until quite recently. This figure was obtained by starting with 2500 BC, an approximate and reasonably assumed date obtained by dead reckoning with the help of written documents. To this was added from fifteen hundred to two thousand years, a time span large enough to account for the stratigraphic accumulation of all the earlier cultural remains down to virgin soil, that is, right down to the beginning of human habitation in Sumer. At that time, it was generally assumed, Sumer was a vast swampy marsh broken up here and there by low islands of alluvial land built up by the gradual deposit of silt carried by the Tigris, Euphrates and Karun rivers...It is reasonably certain that the first settlers in Sumer were not Sumerians. The pertinent evidence derives not from archaeological or anthropological sources, which are rather ambiguous and inconclusive on this matter, but from linguistics. The name of Sumer's two life-giving rivers, the Tigris and Euphrates, or idiglat and buranum as they read in cuneiform, are not Sumerian words. Nor are the names of Sumer's most important urban centers -Eridu, Ur, Larsa, Isin, Adab, Kullab, Lagash, Nippur, Kish- words which have a satisfactory Sumerian etymology. Both the rivers and cities, or rather the villages which later became cities, must have been named by a people that did not speak the Sumerian language, just as for example, such names as Mississippi, Connecticut, Massachusetts and Dakota indicate that the first inhabitants of the United States did not speak the English language.
The name of these pre-Sumerian settlers of Sumer is of course unknown. They lived long before writing was invented. Nor can we identify them from the Sumerian documents of a later day...But this we do know with a fair degree of certainty: they were the first important civilizing force in ancient Sumer, its first farmers, cultivators, cattle raisers, and fishermen; its first weavers, leatherworkers, carpenters, smiths, potters, and masons.
Once again it was linguistic analysis that provided the proof. In a paper published in 1944...Benno Landsberger...analyzed a a number of culturally significant "Sumerian" words -that is, words known from Sumerian documents of the third millennium BC and therefore generally assumed to be Sumerian- and showed that there is good reason to believe that they are not Sumerian at all. All these words consisted of two or more syllables -in Sumerian, the majority of roots are monosyllabic- and in general showed the same pattern as the words for Tigris, Euphrates, and the non-Sumerian city names; Landsberger concluded that they must therefore belong to a language spoken by the same pre-Sumerian people that had named Sumer's two rivers and most of its cities. Among these words were those for farmer (engar), herdsmen (udul), and fisherman (shuhudak), plow (apin) and furrow (aspin), palm (nimbar) and date (sulumb), metalworker (tibira) and smith (simug), carpenter (nangar) and basketmaker (addub), weaver (ishbar) and leatherworker (ashgab), potter (pahar), mason (shidim), and perhaps even merchant (damgar)...It therefore follows that the basic agricultural techniques and industrial skills were first introduced in Sumer not by Sumerians but by their nameless predecessors. Landsberger called this people Proto-Euphrateans...
In archaeology, the Proto-Euphrateans are known as the Ubaid people, that is, the people responsible for the cultural remains first unearthed in the tell known as al-Ubaid not far from Ur and later in the very lowest levels of a number of tells throughout ancient Sumer. These remains consisted of stone implements, such as hoes, adzes, querns, pounders, and knives, and of clay artifacts, such as sickles, bricks, loom weights, spindle whorls, figurines, as well as a distinctive and characteristic type of painted pottery. As already gathered from the linguistic evidence, therefore, the Proto-Euphrateans or Ubaidains, were enterprising agriculturalists who founded a number of villages and towns throughout the land and developed a rural economy of considerable wealth and stability.
The Ubaidians,however, did not long remain the sole and dominant power in ancient Sumer. Immediately to the west of Sumer lies the Syrian desert and the Arabian peninsula, the home of Semitic nomads from time immemorial. As the Ubaidain settlers thrived and prospered, some of these Semitic hordes began to infiltrate their settlements both as peaceful immigrants and as warlike conquerors...it is highly probable that heSumerians themselves did not arrive in Sumer until sometime in the second half of the fourth millennium BC." (pp.40-42. "History: Heroes, Kings, and Ensi's." (Samuel Noah Kramer. The Sumerians, Their History, Culture, and Character. Chicago. The University of Chicago Press. 1963. ISBN 0-226-45237-9 paperback)