As the topic header says, do you?
I have mainly Casio G-Shocks so it’s quite easy for me to clean these. My cleaning method is pretty simple - lukewarm water and a mild soap and I just use my fingers to rub over the bezel and the band. And then of course, a tap water rinse. My reason for doing so is to clear the dead skin cells and sweat marks. One oft spends good money on watches, so my feeling is to keep ‘em looking good.
I suppose with less water resistant watches and those on leather or material bands, this sort of cleaning would be impossible.
Do you have other methods of watch cleaning?