Brendan Perry, formerly of Dead Can Dance put out an album a couple years ago called "Songs of Disenchantment: Music from the Greek Underground " that was a bunch of old Greek folk songs. Probably not as ancient as you're looking for, but not really modern either.

Found a guy on Youtube a few years back named Peter Pringle who plays a variety of ancient music, though it mostly tends to lean Babylonian or Sumerian https://www.youtube.com/@copperleaves

https://youtube.com/@faryafaraji

I think I've heard good things about this person :)

There is no writren record nor recordings (of course) of such ancient music. Yes, there are some hints like greek musical modes but they are not trustable in terms of historicity. You could guess greek folk music has a relationship with this ancient music and speculate a little based on archaelogical evidence of instruments, but: nobody knows how roman or greek ancient music sounded.

Daemonia Nymphe! Love them.

https://www.youtube.com/@the_miracle_aligner

Makes covers in ancient Greek and Latin, among others.

Petros Tabouris - Ancient Greek Musical Instruments

No suggestion other than to listen to some Zorba!