Here is a list of some places to enjoy dining al fresco. Lots of others have at least a little outside dining - if you want to sit outside make sure you ask when you make your reservation.
Maya Beach Bistro @ Maya Beach Hotel
Danube, Austrian Restaurant (screened porch)
Pirate's Grill @ Lost In Paradise Beach Shop
DeTach
Barefoot Beach Bar
Cozy Corners
Yoli's
Radiance's
Gloria's
Pardise Vacation Hotel Resturant and Bar
The Galley
The Shak
The Beach Bar at Turtle Inn
Roberts Grove patio section
Habaneros at Roberts Grove
Bella Maya Pool Bar
Placencia Hotel restaurant on the pier
Laru Beya Restaurant and Bar
The Bonefish Grille @ Singing Sands Inn