There is a plethora of various Middle Eastern art that follows, which I haven't studied too in-depth at all, unfortunately. So the following is a shamefully generalized overview of thousands of years of history, wars, civilizations, and cultures. Here are some highlights.
The Sumerians thrived before 4,500B.C., in Sumer (southern Italy). The Sumerians are credited with inventing the first written language: cuneiform, which uses wedge-shaped characters. The most famous work of literature written in cuneiform is the Epic of Gilgamesh, which can be found at your local Barnes & Noble booksellers store (I love product placement). Sumerian art includes statues that were crafted and left for religious worship, as offerings to their gods. Ziggurats, which were stepped mountains made of dirt, also became popular during the Sumerian civilization, and there are several still standing today.
The Sumerians were succeeded by the Akkadians, a Semitic people, the king of whom, by 2,340B.C., controlled Sumer (the entire region between the Mediterranean and the Persian Gulf). From the Akkadian empire we see art like the Stele of Naram-Sin, a limestone relief commemorating a military victory. Hierarchic scaling is the feature element here, as the gigantic-looking Akkadian king literally tramples over the dwarf-like, conquered bodies of his enemies.
A time of Neo-Sumerian civilization followed, which was then succeeded by the Babylonians. The Tower of Babel is estimated by archaeologists to have been in construction before c. 3,500B.C., and King Hammurabi is believed to have ruled around 1,800B.C. The Babylonians, too, made steles, and much of their art was based on punishment for certain crimes, since Hammurabi was famous for his Code of Laws, which was a series of 282 laws inscribed in a stele in cuneiform. The code stresses punishment for violating the laws, and the punishments are not at all flippant or superficial (i.e., an eye for an eye).