From my study on the subject, I got an impression that is was similar to the Nile banks and Delta. There are references to it being both fertile plains as well as marshland. The produce grown in this area included wheat, sesame, palms and apples.

The majority of the larger cities were situated between the Tigris and the Euphrates, or on the East bank of the Tigris, where water was in great supply. Town and cities on the West of the Euphrates tended to rely on water from tributaries.

There was also canals and irrigation, extending the value of the rivers to terrain further inland. There are references in Herodotus (1.193) and Pliny the Elder's Natural Histories of crops being cultivated 2-3 times per year, largely due to the consistently hot weather and the irrigation providing sufficient water supply.

The Nile is a useful comparison in the sense that there's a strong division between the cultivated realm, the realm of cities, orchards, and farmland(with orchards usually close by or even in the city and farmland further out) and steppe. Another useful comparison might be the Mississippi for two reasons. First, the coastline was originally far closer to the shoreline than it is today; cities like Eridu and Uruk were originally within a relatively short distance of the coast and texts like Adapa or even Gilgamesh describe a close relationship between these cities and the sea. So we might imagine the cities of the southern half of Sumer especially (the cities of say, Eridu, Uruk, and Ur) as being in earlier periods especially somewhat like the cities of the Mississippi delta, where there is a close co-existence between city, sea, and marshy littoral. Even inland cities were usually on a canal or riverbank of some kind. If we look at this map of Nippur for instance we can see where it was formerly bounded by canals(http://oi.uchicago.edu/research/projects/nip/nsc.html).

Second, both are very much man-modified landscapes. The Mississippi is heavily diked and channelized for irrigation and flood control purposes. Likewise, the landscape of Babylonia and especially southern Babylonia was heavily crisscrossed by various river channels and canals that were built to supply irrigation water. Rather than one channel and fields extending along that channel to a certain depth, we might rather think of capillaries of canals extending over the landscape(many agricultural land sale contracts specific a water boundary that is either a river or canal), the canals visibly extending several meters above the surrounding plain(southern Mesopotamia is an extremely flat place compared to Northern Mesopotamia, and the Euphrates especially is accordingly flatter and slower in that region). These canals were used for irrigation; they could also be used for transportation and shipping of goods to supplement the limited agricultural productivity. At any rate, the canal system was of great importance both to the landscape and agricultural productivity of Southern Mesopotamia, even if the impossibility of non-irrigated farming has been overstated. The climate also fluctuated slightly but noticeably; for instance it has been proposed that the Neo-Babylonian period was one of increased rainfall and longer growing seasons compared to the Old-Babylonian period. Thus agriculture may have been easier or more difficult depending on the time-period. I have also seen the suggestion that the fields were structured by water use intensity, with orchards and gardens closer to the levee top and barley fields farther away.

Remaining oak, cedar and pine forests in Lebanon (and probably elsewhere in the Middle East, but I guess that it might be safer as a travel advice than Syria) can give you an idea of the general appearance of what vegetation looked like in this area — in fact, Lebanon provided the very literary archetype of the forest. Unfortunately, surviving fragments of woodland are rather remote and come mostly by patches, which means that the feeling of awe which must have stricken travellers in the area is hard to imagine for the modern tourist.