How would the Greeks have come into contact with influences from the Near East? Identify and explain as many means of this cultural transmission as possible.

Travel by land and sea around the Mediterranean has been ongoing for many centuries. With technological advances that improved the seagoing properties of ships and enabled better road construction, connections became easier and more frequent. The desire for exotic goods and superior materials that increased as Greek civilization developed in turn brought increased economic interactions.
War and hostilities were a main source of interaction. The inevitable political conflicts and movements toward conquest led to military interactions as well. Under the rule of Darius (who ruled 522–486 BCE), for example, Persia expanded into mainland Europe, with particular interest in Athens. As Darius demanded Greek submission, the Athenians and Spartans joined forces to resist Persian control, to which Darius responded with massive naval power directed at the Cyclades. Under Xerxes, the military efforts at political expansion continued. The consolidation of northern Greek power under Philip II of Macedon in the 4th century BCE provided the foundation for his son, Alexander the Great, to turn the tables and expand into Persia.
https://www.ancient.eu/Persian_Wars/

https://www.britannica.com/biography/Philip-II-king-of-Macedonia