Archaeologist here. The influence of Near Eastern culture on the western Mediterranean was huge. Even in the Mediterranean Bronze Age (ca. 3000 - 1100 BC), the Minoan culture on Crete (ca. 2600 - 1400 bc) and the Mycenean culture (1600-1100 BC) used a same sort of early political system that was used in the Near East as well: the Palace system. This meant that there were "palaces" that worked as big redistribution centres in the region they were located. Palaces are found in Mycenaea on the Peloponnese, Knossos on Crete and a whole lot of other places. Later, during the Iron Age, the eastern influence was again huge. The Greeks adopted the Phoenician alphabet and made their own version of it, suitable for the ancient Greek language. During the Iron Age (and even before) in Italy, before the Roman conquest, cultures were very much influenced by the east. The Etruscans for example used a lot of eastern influence in their art and incorporated Greek gods in their religion. You should look up the Regolini-Galassi grave, a great Etruscan example of how much they were influenced by the east. A huge amount of pure golden objects have been found, beautifully decorated with eastern-like art. The main reason for the spread of eastern influence are indeed the Phoenicians, who traded everywhere in the Mediterranean region.

There is also evidence for a connection in terms of mythology and poetry, at least between the Levant/Anatolia and the early Greek authors and of course particularly in the fields of science and astronomy.

And writing (first the concept per se and then the alphabet) is probably the most important innovation spreading from east to west.

A great resource is M. L. West: The East Face of Helicon. West Asiatic Elements in Poetry and Myth (Oxford:Clarendon, 1997)