Hmm I don't have sweaty wrists :O But to clean in general the soft inside bit what do you use? Just leather wipes?
You'll be surprised at how it goes over time. Leather wipes if you can but try not to use anything too soapy because otherwise you'll find that some of the dye in the leather gets drawn out (that is if the underside of your strap is dyed of course).
I got my first leather strap watch for my 21st and after a year it started to smell a bit because I didn't take care of it from the off. Some sites recommended just a quick clean with a damp cloth.