Looks like you are asking more about ethnic groups or languages rather than civilizations.
BTW - are you also interested in Anatolia / Asia Minor, or only Europe up to Bosphorus?
Here are some confirmed and hypothetical, major ethno-linguistic groups in Ancient Europe:
(Some of) Indo-European groups (incl. hypothetical ones):
Italic peoples
Celtic peoples
Ligurian peoples +
Hellenic peoples
Phrygian peoples +
Dacian peoples +
Thracian peoples +
Germanic peoples
Baltic peoples
Slavic peoples
Iranian peoples
Armenian peoples
Venedic peoples +
Illyrian peoples + (?)
Lusitanian peoples +
Sorothaptic peoples +
(Some of) Non-Indo-European groups with known names:
Phoenician peoples +
Iberian peoples +
Vasconic peoples
Tyrsenian peoples +
Pelasgian peoples +
Tartessian peoples +
Aquitanian peoples +
Turdetanian peoples +
Finnic peoples
Sicilian peoples +
Minoan peoples +
Kartvelian peoples
Caucasic peoples
Sign "+" placed after a group's name means that their languages, cultures, are extinct today.
In case of Illyrian languages some think that Albanian is descended from one of them (if true it means they aren't extinct), but others disagree.
Kyriakos said:
the first Greek civ was the Cycladic one, the Minoans came at least half a millenium later and then there were the Myceneans.
The Minoans were not even Indo-European, so how could they be Hellenic or Greek? The Mycenaeans were the first Greek-speaking civilization.
Saying that the Minoans were Greek is like saying that the Byzantines were Turkish. The same area doesn't equal the same culture & language.
sydhe said:
If you go a few centuries further, you get the Myceneans and Etruscans.
Etruscans were also Non-Indo-European. Etruscan is classified as one of Tyrsenian languages (see my probably incomplete list above).
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BTW - is the name of Pre-Celtic (Non-IE) inhabitants of Britain known or unknown?
I have on my mind that I have heard this name somewhere, but I'm not sure.
Maybe such a name survived in Insular Celtic mythology and legends?