According to the Norwegian meterological institute, the answer is that it is best to run. According to Mythbusters (quoted in the comments to that article), the answer is that it is best to walk.
My guess would be that this is something that can be properly modelled mathematically and solved. I suspect that I'm not alone in making a start on this, but as it's usually during a rainstorm that I think of it, I don't get far before deciding that the best strategy is to call a taxi!
I would also guess that someone has already done this and figured out the answer. If I'm right about that, I would like to read about it and learn what the real answer is. Thus my question is most likely: can someone provide a reference for this?
(If I'm wrong that it's already been solved, then some pointers as to how to solve it or even an explanation of what the main issues are, would be most interesting.)
As well as simply satisfying my curiosity, this is also one of those questions that non-mathematicians often ask (or at least, that non-mathematicians can easily understand the motivation for the question) so if there's a reasonable answer then it would be a good way to show off some mathematics.
Update This question has a long and curious history. There's a discussion about it at the meta site: http://meta.mathoverflow.net/discussion/90/question-about-walking-in-the-rain