You want an island that’s a little out of the way of mass tourism, so you can see some local music and culture. Or an island that’s big enough that not every part of it is soaked with mass tourism.
You won’t see much, this is 2016 after all. And as I posted here (Nick Nicholas’ answer to What should I know (but don’t) about the culture and history of the Cyclades in general and Syros in particular?), the indigenous dialect of the Cyclades (the most archetypal Greek islands) died out a century ago.
I actually haven’t done the Greek islands, partly because that’s exactly the kind of thing I’d be looking for, myself. I’ll make a wild guess:
- Of the Ionian islands, not Corfu, probably not Zante. Maybe Cephalonia.
- Of the Northern Aegean, Thasos and Samothrace definitely (but why would a tourist want to? 🙂 , likely Lemnos. Lesbos is big enough that there are traditional bits left, and even visible differences within the island (I’ve been there), but you have to hunt them down.
- I don’t even think of Euboea as an island, but: Euboea.
- Of the Cyclades, I’d go with any island you haven’t heard of. Folegandros, maybe, or Amorgos. But I’m guessing here.
- Of the Dodecanese, I suspect all of them; 11 of the 12 are tiny and out of the way, and the 12th, Rhodes, is big enough that something indigenous will have survived the tourists.
- Crete. Of course. Not the coast around Iraklion. Ever. But the hinterland? Yes.