Instructions
Send a message to all of the people on your Xbox LIVE friends list. Ask them to report your gamertag as being offensive. Your gamertag doesn't actually have to be offensive as long as enough people file a report. Microsoft will allow you to change your gamertag without charging any fees after you have been reported multiple times.
Join any game and use the voice-chat feature when the game-play isn't too intense and ask other players to report your gamertag as well. Not everyone will comply but many gamers have had a similar problem as yours and may be willing to help you out.
Use online forums to ask other gamers for help if you haven't had any luck by contacting your friends and have received no help from random players. Make a post on an Xbox-oriented forum, post your gamertag and ask other players to report your gamertag very politely. The exact number of reports you have to accumulate before Microsoft will allow you to change is unknown, but it should take less than two hours to complete the process.