How do you clean a felt white cowboy hat?

It's probably best to hold the hat with four fingers of your hand inside the crown and your thumb on the edge of the brim, steadying the hat. I get loose dirt off with a soft cosmetic brush (of the type the ladies use to put on blusher and such-like). The nap of the felt on the crown and up-side of the brim goes counter-clockwise, so brush in that direction. Once the loose dirt is off, use a dry sponge to go over the crown and brim again. This also helps raise the nap. Actually, you might want to go to your local western-wear store and pick up a little orange sponge specially-made for cleaning hats by M&F Western Products, out of Sulpher Springs, Texas. (No, I don't work for them or own stock in the company. It's just a good little product, and very reasonably priced.) Set the hat on its crown and repeat the brush-and-sponge routine on the under-side of the brim, but this time you're going clockwise! This is good for routine, day-in, day-out care. At some point, though, you're probably going to need to take it to a hatter for real deep-cleaning. Good luck.