I've heard that since the water going down the drain spins clockwise in the northern hemisphere, then the water going down the drain should spin counterclockwise in the southern hemisphere. Is this true?

NO

It's just that simple. It's a popular misconception that refuses to die. Don't give it another thought.

Minnesota is correct. This doesn't happen. Besides the physics behind it, my kitchen and bathroom sinks (only feet away from each other and well within the same hemisphere) drain in opposite directions.

In case you're curious, however, here's how the myth started. The first thing is that toilets do always drain in a particular direction... but this is dependant on the design of the toilet, not the hemisphere. The spin gives a bit of an extra kick so that we don't need to use as much water.
Weather formations, especially hurricanes, do spin counterclockwise in the northern hemisphere and clockwise in the southern hemisphere. This is due to something called the coriolis effect which basically occurs because the surface of the earth moves faster closer to the equator (because it is wider there ... think of it as a record with the outer edge being the equator, and some point closer to the center being where you are... he edge has to move faster than where you are because it is a longer path and makes the same number of rotations per minute). The coriolis effect is very weak, but gets stronger the larger the thing it's affecting it, and it takes a while to build up noticeable speed. So, it does affect huge things that stick around for a while, like hurricanes, but doesn't have any noticeable effect on anything as small and short-lived as the draining of a sink.

Hopefully that explanation was easy to follow, but the take-hom message is NO, this will not affect your sink despite the number of people and publications that have fallen for this myth.

here's a good page to look at for more info:
http://www.ems.psu.edu/~fraser/Bad/BadCoriolis.html
http://www.windpower.dk/tour/wres/coriolis.htm

Thanks so much to Methos5000 and Minnesota. Methos5000, your links and explanation were very thorough and easy to follow. Much appreciation!

There was a show called Mr. Wizard's World in which Don Herbert demonstrated this effect on a small merry-go-round. He had a tub of water with a plugged hole at the bottom. He positioned it and himself on the merry-go-round about two feet from the axis and had the children spin him around briskly. When the plug was pulled the draining water did rotate in different directions, depending on the merry-go-round's direction. This demonstrates that the coriolis effect is only observable with small amounts of matter in a children's playground.