Yes, recap should do it alright. I had exactly the same problem (checkerboard screen) with my Classic II and the board showed absolutely no signs of leakage whatsoever. Actually the board looked brand new! Still though the caps must have dried out or something because recapping fixed it.

Strangely enough, after washing the board it caused a short somewhere so that the power suppy wouldn't even start up (click click sound). Maybe some water got into one of the faulty caps and caused the short. I wasn't expecting anything when I recapped it so now I'm even happier!