I really have had much success with tea balls. I get the loose tea leaves at the bottom of my cup too. <img src="/images/graemlins/tongue.gif" alt="" /> I use a small square of cheese cloth and 100% cotton string. Sounds kinda weird, I know, but it works.
You just put your tea in the middle of cheese cloth, close it up and tie it off with the string. Pour boiling water over the little bag and let it steep. When you're done, empty out the old tea leaves and hand wash the square. I use a lemon and vinegar mix to wash the cloth and then let it air dry.