Here is a home remedy that someone told me about for clearing blocked drains. [I thought long and hard about how to phrase that sentence, given this crowd!] In any case, it is supposed to be less harsh than Drano or Liquid Plummer, and a lot cheaper, too.
Empty approximately 1/3 cup of baking soda into the drain, and follow that with 3/4+ cup of white vinegar. It will fizzle and bubble. Let sit about 10 minutes, then pour a bunch of boiling water down the drain. Repeat as needed.