In reply to damowilk:
A lot of people seem to believe this and I'd been wondering myself where the belief came from. I climbed with a friend who insisted on climbing the clipped side because even though he found it really irritating, he had a feeling it was safer, though neither of us could think of a reason why.
It's certainly true that you need to climb on the clipped side of the rope on any sufficiently overhung or traversing route. Apart from that, though, it seems more like a matter of personal taste to me. The captive rope could be annoying while climbing but then having to unclip can be irritating too.
As you say, for straight up routes, the best option is probably just to unclip on the way down. If, for whatever reason, I haven't done that on my way down, my partner normally climbs on the unclipped side of the rope because she finds it less irritating.
The only other arguments I can think of are wear on the draws or mobility of the belayer. Perhaps indoor walls promote always climbing on the clipped side to avoid people with no common sense climbing the unclipped rope on overhangs and/or to slightly reduce the wear on their draws (particularly the first).