As OP clearly knows (by his “pre-Persian” restriction), the main Semitic name for Greeks, Yunan, derives from Persian contact with Ionian Greeks.
We know that the Hittites used the term Achiyawa to refer to what we reasonably guess were the Achaeans; that’s contact dating from Mycenaean times.
From Greek Contact with the Levant and Mesopotamia in the first half of the first millennium BC: a view from the East by Amelie Kuhrt, University College London – Academia.edu, it looks like Assyrians in the 7th century BC were already referring to Ionians. (See also Ionians.) I’d be surprised if the Babylonians knew the Greeks at all before then.
The really interesting question is what did the Phoenecians call the Greeks, before the Ionians settled Asia Minor. I’m not finding it online, and you know, there may well not have been an established term. If the Phoenecians had one, I’d have thought it’d have shown up in Wikipedia.