I believe that some of you here were misled to believe that T was high in south-western Iran because of the mistaken map of T on Wikipedia, which is based on the K in Nasidze et al. 2004. The only reliable study for hg T in Iran is the more recent
Grugni et al. 2012, which found mostly haplogroup L in southern Iran, and
no T at all in Khuzestan and Isfahan.
Considering that there is hardly any T in the region of Elam today, and that the most common haplogroups are J2a, J1 and R1a, I would rather think that the Elamites were predominantly J2a (R1a being Indo-Iranian and J1 probably Arabic). Besides, the influence of Elam was directed eastward towards Baluchistan, where once again J2a dominates.
Haplogroup T is found mostly in Mesopotamia, then in the northern periphery where the Sumerians had colonies (Syria, eastern Anatolia, southern Caucasus). Its low frequency in historical Sumer (Iraqi marshes) is mainly due to the extremely strong presence of Arabs (74% of J1-P58), who arrived relatively recently. Additionally, that study only tested Arabs from the Al-Hawizah marshes, along the Iranian border, not between the Tigris and Euphrates. If we take out all the clearly post-Chalcolithic arrivals (J1-P58, Q, R1a and R1b), what is left in southern Iraq is J1(xP58), E1b1b, J2a, T, G and L, in that order of frequency. The
Kuwait Y-DNA Project can also serve as a comparison. It shows that hg T and J2a are the two most common in the region after the Semitic J1-P58 and E1b1b.
The difficulty is to determine whether J1(xP58) and T were brought by Semitic people (Akkadians, Babylonians, Assyrians, Arabs) or were already there before. E1b1b is the Afro-Asiatic haplogroup
par excellence, so I would not consider it native Sumerian. J2a could be Sumerian, but it has very little diversity among southern Iraqi (especially compared to the rest of Iraq) and might well have come with the Arabs or other recent invaders.
That leaves G, E1b1b, J1(xP58) and T as the possible Sumerian+Akkadian haplogroups. I don't have any details about the subclades of haplogroup G found in Iraq, so I cannot judge when it could have arrived.