For most of antiquity there was an indigenous population called the Pelasgians living in Greek speaking lands that was already then understood to predate the Greeks.

Italy had a lot of different peoples living in it before they melded into a general Roman identity during the centuries, the most famous and probably most significant being the Etruscans. The Romans themselves had a national mythology of being descendants from the Trojans, but I don't know how seriously they took it.