Chiang Mai:
Galae, a Thai restaurant part-way up the mountain. They have a roped-off table reserved for the King.

Mozart, a nice European street restaurant at the Night Market.

Both of the above are pretty reasonable.

Phuket:
All in Patong---

Very expensive:
Fisherman... wonderful outdoor seafood restaurant at the Novotel

Ban Rim Pha... Thai restaurant on the beach, great atmosphere.

Inexpensive:
Rondtarn... Steak restaurant off Soi Bang La, the main shopping street. It is acually on Soi Patong Resort.

In Bangkok, I would personally avoid the overpriced and unremarkable Oriental Hotel, although some people love it. The buffet at Sukkothai Hotel is the best.

Have fun!