Norwegian Cruise Line.

Norwegian sounds like what you want- if you are thinking of cruising in the summer take a look at the brand new NCL Epic, she looks wonderful. During the summer the choice of NCL ships is reduced.
RCCL also has 'My time dining' which gives you the choice of eating on your own schedule during dining room hours, they have several ships sailing the Caribbean year round, from various Florida ports.

Caribbean January 2010 thinking of