A lot of pre-Greek substratum is non-Indo-European, and came from some language family/ies that went extinct before any of it was recorded.
But in Asia Minor the substratum was the Anatolian language group, Indo-European relatives of Greek. Some say “Indo-Hittite” instead, to put the Anatolian languages on the same taxonomic level as Indo-European. In other words, the first split of Indo-Hittite divided it into Indo-European and Anatolian. Anatolian languages include Hittite, Lydian, Luwian, Lycian, and Carian. The name Asia itself comes from the Anatolian name Assuwa, which was the northwest of Asia Minor and many centuries later became the Roman province of Asia. A number of Greek mythological names originated in Anatolian languages, including Apollo, Aphrodite, Artemis, Cybele, and Troy.
I am more persuaded by the theory that the Minoan language written in Linear A was of the Anatolian family, probably a form of Bronze-Age Luwian. It was also once proposed that it was a form of Semitic, from Syria. The scholarly consensus today is that Minoan and its descendant Eteocretan are still unclassified.
One other Pelasgian language has a known affiliation: the Lemnian language once spoken on the Aegean island of Lemnos (just northwest of Lesbos) was related to Etruscan. Together they make what’s known as the Tyrrhenian (some call it Tyrsenian) language family. The pre-Greek language of Cyprus, Eteocypriot, is also unclassified but it may be related to Tyrrhenian or maybe to Northwest Semitic via Syria. The Etruscans originated from Anatolia and were part of the migrations of a number of peoples in and around Asia Minor during the late Bronze Age, many of them westward to Italy.
The Elymians who settled in western Sicily likewise originated from Asia Minor. The Sicels, the pre-Greek nation in eastern Sicily, had immigrated there from Italy and spoke an Italic language related to Latin.
The pre-Indo-European substrate in Anatolia, before the Hittites appeared, was Hattic, which may be related to the Northwest Caucasian languages.