That day I was telling my cousin about this dilemma and she made a wonderful suggestion to me. Use a pipe cleaner to get down in there. I happened to have some pipe cleaners on hand from a craft project so I decided to give it a try. Sure enough, I was able to scrub the inside of the straw and mouth piece.
While the pipe cleaner works very well, there are times when I notice something in the straw or mouth piece that I cannot get out with the pipe cleaner no matter how hard I try. Then I discovered for tougher tasks, using the thinner end of a chopstick works great.
So if you're having troubles cleaning your child's FUNtainer, invest in some pipe cleaners and chopsticks!