Our son was recently fitted with a bite plane to correct his overbite. It is similar to a retainer and is removable. The first thing we noticed it that it gets stinky pretty quickly. Simply brushing it with toothpaste did not get rid of the smell.
We have found that brushing it with toothpaste, then soaking it in denture cleaner for a few minutes works really well to eliminate odors. In addition, it is important that you take extra measures to clean your mouth, as the odors build up there too. Our son brushes his teeth really well, especially the backside of his teeth where the bite plane traps food. He also brushes the roof of his mouth, then follows that with mouthwash. I highly recommend doing this at least 2-3 times a day.
This process seems to do the trick! Anything short of this and his breath gets pretty funky. :)