Did they ever! The Mesopotamian pantheon is huge and quite abundantly documented, and we have mythical texts from Mesopotamia, Anatolia, Ugarit, Egypt, and the later periods of pre-Islamic Persia, as well as a wider range of inscriptions and god lists(although our picture of the pre-Islamic Arabian pantheon is largely limited to inscriptions). A good introduction to the topic is Stephanie Dalley's Myths from Mesopotamia; in addition quite a few Sumerian mythical texts are available online for free via the Electronic Text Corpus of Sumerian Literature (http://etcsl.orinst.ox.ac.uk/). Hittite mythological texts are collected in Harry Hoffner's Hittite Myths. I'm not aware of a stand-alone volume on pre-Islamic Arabian myth and pantheon but Robert Hoyland's Arabia and the Arabs should be a useful starting point for your reading there. One thing you should be aware of is that there were many ancient middle eastern pantheons, not one-even within the same culture or more precisely region and linguistic sphere the mythology and pantheon changed a good deal over time and from place to place because there was less of a product of codifying myth comparable to that which took place in the Classical world.

EDIT: Of course, feel free to ask any more detailed/followup questions.