We found this wonderful Airbnb place (Konstantinos' 'beautiful house above the sea') in Mandraki, cosy and close to the main square.
The kind owners at the beautiful Villa Kalypso allowed me to bring Lisa - I mostly kept her on the balcony or the terrace.
Anemos Guesthouse in Olympos village (see www.visitolympos.com), and Anemos Sunrise Taverna on Agios Minas beach (call 0030-2241500377).
Restaurants that have welcomed Lisa include Faros and Nautilos in Livadia, and Elpida and Delfini in Ayios Antonis. In summer, Eristos beach and the kantina there are dog-friendly! And dogs are welcome at Antonis' Apartments in Ayios Antonis.
RHODES: The Hotel Congo is so dog-friendly that they remember the dogs' names better than the owners! An absolute delight to stay there, and there's a bar and restaurant. It's close to Zephyros beach, about a fifteen-minute walk from where the big Blue Star ferries dock. The Olympos Pension in Rhodes Old Town also has some dog-friendly apartments in the winter.
Lisa got an extremely warm welcome at La Boheme, a bar on Theodorakis Street in Mandraki, near the bus station and New Market. Lovely place for humans, too... And along the road are Perfect Spot Apartments, where the top floor suite has a huge terrace for a dog to run around on.
FERRIES: In my experience, the staff are very happy to accommodate dogs on the deck, and big long-distance ferries tend to have boxes where you can leave your dog - I'd bring a blanket or bed though. The Dodecanese Express/Pride have started requiring dogs to wear a muzzle.
CRETE: When I visited Sitia, I was welcome to stay with Lisa at the excellent and central Hotel Itanos. This was organised for me by Minoan Car Rental.
SYMI: The stunning Old Markets hotel on the harbour accepts canine guests in their largest suite. The friendly owner sent me this, which they found online: