According to the Telegraph these are:

The Old Markets, Symi

A two-hour ferry ride from Rhodes,   the Dodecanese island of Symi is so lovely you don’t really want to tell  others about it. A rugged gem with a harbour of crumbling neoclassical   mansions, it offers a wonderful local experience and some of the best food   on the islands. Until recently there were almost no good hotels on Symi, but   now there is an outstanding one: the Old Markets, housed in a restored   market building where sponges and spices were traded until it was bombed in   1944. Located up about 50 steps in the harbour area, the Old Markets is   owned and run by British-born Andrew Davies and his Italian partner, Carlo   Palumbo. Each room is inspired by a trade route and filled with antiques and   contemporary textiles. Four are off a terrace overlooking the harbour, the   perfect spot for a champagne breakfast, and the fifth, a suite, has a   private terrace. Andrew and Carlo are professional chefs, so the food is   great. Lunch and dinner are on request, and at dusk aperitivos are served on   the terrace.

Details: Doubles from €175 per night. Children must be 16 or over unless  you book the whole place on an exclusive-use basis (i-escape.com/the-old-markets).   

The Peligoni Club, Zakynthos

We love the Peligoni Club, a rustic, chic retreat on a sleepy hillside on the   north-eastern tip of the Ionian island of Zakynthos. It is run by the highly   dedicated Shearers, a British family who bought the place in 2005. Apart   from the genuinely warm, personalised service, it has a unique set-up: the   central club has a restaurant (the fresh local food is very good), a bar,   tennis courts, a swimming pool, the one-room Goat Shed spa, and a series of   glamorous sunbathing terraces descending to the crystal-clear sea. Then   there are the stylish cottages, villas and rooms, scattered about the   surrounding hills, the furthest a 15-minute drive away. There is everything   from a restored olive grower’s cottage for two to the super-luxe Figari   villa, sleeping 18. We stayed in the secluded Anna’s House, with a heavenly   terrace and pool.

Details: The Peligoni Club offers a range of accommodation sleeping two   to 20. Anna’s House, which sleeps eight, costs from £3,500 per week. Club   membership, including use of all watersports/sailing facilities, costs from   £110 per person (020-8740 3001; peligoni.com).

Santa Marina, Mykonos

Mykonos   may not be the obvious choice for a family holiday, but Santa Marina   Resort & Villas, which overlooks the Aegean sea on a secluded peninsula   in Ornos Bay, is up there with the best Greek family resorts. If you care   about this sort of thing, Madonna, Tom Hanks and Sarah Jessica Parker are   just three of the celebrities to have decamped here with their broods, all   attracted by its tucked-away oceanfront location (just 3km from Mykonos   town), its laid-back exclusivity, its excellent facilities – including lots   for kids – and its sandy beach. The resort has just refurbished all of its   bedrooms, giving them a breezy New England feel, and made other tweaks, like   adding the Bayview restaurant, which serves delicious Mediterranean food and   has a Japanese chef overseeing a sushi bar. There’s also the Lotus Spa,   which uses products by sustainable beauty brand Ila, four swimming pools, a   playground and a supervised kid’s club.

Details: The resort is seasonal and opens from May to early October each   year. Sea-view doubles from €240 per night, including breakfast. Book online   now for early-bird savings (0030-210 4226571; santa-marina.gr).  

Onar, Andros

It’s a bumpy ride down a winding track to Onar (‘dream’), a retreat within 18   acres of protected wetland overlooking the amazing Ahla Beach on the eastern   coast of Andros, but it is worth it. The owner, Mateo Pantzopoulos, first   discovered the spot when he was 15, immediately returning with a tent. In   2005 he launched Onar with five boho-chic electricity-free houses, all a   short stroll through the valley to the beach. Back then it was really just a   secret place for Pantzopoulos to take friends. Today there are 10 rustic   stone cottages, all with kitchens, living rooms, fireplaces, shaded terraces   with hammocks, and home comforts such as air conditioning and Wi-Fi. Meals   are served communally, around a big table under a tree in the garden, or in   the privacy of one’s own cottage. The Michel Roux-trained chef Ilias   Kariotoglou joined the kitchen last summer, so the food, which has always   been good, is now even better and overspilling with local ingredients, from   feta, honey, olive oil and tomatoes to gloriously fresh seafood.

Details: From €170 per cottage per night (020-3397 0085; chicretreats.com).