If its traditional Christian values your looking for, for yourself, or or your family, then consider the following islands which are less traveled by foreigners and consequently less influenced by them and more traditionally Greek.
Generally every Greek island with a substantial year long population has family atmosphere. You just have to seek it out and be patient.
Greeks have families like everyone else, they raise children, send them to school, work, shop and go to Church.
The Church has an incredible amount to do with Greek culture. In fact to be considered a real Greek you have to be Greek Orthodox. - at least according to the Greek Orthodox Church.
Greeks measure the year by religious holiday just like Protestants, Catholics, Muslims and Jews do except in Greece there is a religious holiday everyday.
Such holidays, festivals and weddings are great times to be in a Greek village or small town. Greeks let their guard down at such events and its easier to make friends.
Also, planning your travels in Greece for the off-peak season will make it much easier to get to know the Greek people and to appreciate their strong sense of family. You don't have to come in the dead of winter either. Just avoid July and August.
But what if you are not going to be in Greece on a major Greek religious holiday and don't have any Greek friends getting married? What if you cant help it and the only time you can visit is in late July and August?
Then your best bet is to go to where the real Greeks live. Greeks everywhere like to talk to visitors from abroad and certainly aren't short of opinions and like to voice them. So being sincerely friendly and politely inquisitive is the way to go!
During high season the islands below are all less visited and do not suffer from mass tourism. On them you can have a wonderful time and enjoy the slower pace and with it the greater likelihood of getting to know real Greeks.
A word about Crete: Crete has its own traditions well worth getting to know but avoid Chersonisos and the large tourist enclaves on the islands north coast. Central Crete and the south coast are quietest. A car rental would be very helpful.
Cyclades: Anaphi, Folegandros, Kimolos, Kythnos, Serifos, Sikinos
Dodecanese: Aghios Efstratios, Astypalaea, Chalki, Kalymnos, Karpathos, Kassos, Kastelorizo, Leros, Nisyros, Tilos
Ionian: Andi-kythira, Ithika, Kythira, Lefkada
Sporades: Allonisos, Skyros
North & East Aegean: Chios, Limnos, Samothrace