There was certainly contact between the two regions and striking parallels in their respective mythologies, and literary and artistic traditions which point to Mesopotamian cultures having directly influenced Greek culture. For example Aphrodite is argued to be a version of the Mesopotamian Goddess Ishtar, rather than coming from an Indo-European tradition.

During the Mycean period, which was a particularly formative era for Greek culture; with literature such as the epics of Homer being set at this time, there were Greek settlements as far East as modern day Syria and these had trade and cultural ties to nearby Mesopotamia.

Intensive contact occurred when either the Greek world expanded eastward or when the Mesopotamian world, usually the Assyrians, expanded westward.

There is an important caveat here though - as far as I'm aware where Mesopotamian influence can be detected in Greek culture it is re-figured in a Greek setting without explicit acknowledgement of a non-Greek source. That would point to appropriation/adoption of cultural elements rather than studying them in their original cultural context.

Source: "Greek Myths and Mesopotamia: Parallels and Influence in the Homeric Hymns" ( Charles Penglase) GOOGLE BOOKS LINK