I am currently reading Nicholas Ostler's great book Empires of the Word: A Language History of the World. The third chapter is dedicated to the languages of the early Middle Eastern civilizations. If there is a link at all between languages and genetics, it is among Neolithic tribes and early civilizations that the correlation should be the strongest, before empires imposed their own (written) language on others.
The first written language in Mesopotamia was Sumerian, apparently unrelated to any other known language, except possibly Georgian. If the a relation with Georgian does indeed exist, it could mean that the early Sumerian belonged to haplogroup G. Based on present Y-DNA frequencies in southern Iraq, it would be more likely that J1 was the predominant haplogroup in the Sumerian city-states of Ur, Uruk, Nippur, Eridu, etc.
Sumerian was replaced by Akkadian in Mesopotamia. Akkadian is a Semitic language related to the tribal tongues of the desert nomads from Syria to Palestine. Semitic people and languages are typically associated with haplogroups E1b1b, T, J1 and J2. Akkadian being from northern Mesopotamia, where J2 is prevalent, early Akkadian speakers were likely to belong overwhelmingly to this haplogroup. However, it is likely that the deeper origins of all Afroasiatic languages start exclusively with haplogroup E.
What is more interesting is the case of the Elamites. Their civilization was centered in the west and the southwest of modern-day Iran, and their language related to Dravidian and languages of the Indus civilization. This South Asian origin could correspond to the relatively high frequency of haplogroup L (and minor presence of H) in southern Iran. Furthermore, the Akkadians are known to have deported many of them in various parts of their empire, as far as the Levant. This would explain the presence of hg L at low frequencies in the Middle East, and notably the Levant.
The Medes and Persians are supposed to be the Indo-European invaders who brought R1a (and maybe other haplogroups) into the Middle East.