Schinoussa (Σχοινούσσα)
Very quiet and unspoilt island. Part of the 'Lesser Cyclades' to the south of Naxos. Very much off the usual tourist trail and only 3 square miles in area.
This must be what the Greek Islands looked like 50 years ago. When we visited in September 1999 there were just a handful of other tourists and only a few rooms available to rent. We were easily able to find a beach to ourselves. I imagine it has changed a bit since then but it still has a reputation for being quiet and charming.