If you can find a dry cleaner that will protect the integrity of the tzitzis, this would be the most ideal way to clean them.
Washing them by hand would be the second best. This helps in the following ways: the fabric of the tallis katan could become damaged or at least weakened by the spinning of the machine as well as the tzitziyos becoming tangled.
In addition to washing them by hand I would suggest using a soft detergent specifically designed for a tallis katan (assuming it is wool) such as [Woolite][1]Woolite.
After washing the tallis katan it is best if you can dry it quickly otherwise it will smell badly for some time until it dries. I would suggest hanging it in front of a fan until it is dry. [1]: http://www.woolite.us/