I found this cheese the other night. It was obviously thrown out because of the big patches of mold growing on it, even though the cheese is still good and fresh.
Once it is satisfactorily clean, put it in a clean, dry towel and pat it dry. It is important to make sure it is good and dry before wrapping back up, or it will just get moldy again in a couple days.
Voila! A fresh, clean block of cheese. This pepper jack was also pretty moldy, so I tried washing it too, and it worked just as well. Wrap it up tightly in plastic wrap or keep it in a well fitting tupperware container; the less air its exposed to, the longer it will keep before getting moldy again or drying out.