I had no problems flying Continental to Belize. In fact, I'd rather fly them than any other airline.
The only problem we encountered, was going HOME, we only had a 1 hour layover, and after going through customs we almost missed our flight. Thankfully we ran to the gate & made it just as they were closing.
Lesson learned: Next time, set up for a longer layover when returning into the US.
Thankfully all our luggage was carry on, otherwise we definetly would have missed our flight.