Anthony,

I doubt bleach will give you any long lasting results. The problem is that the water in the Bleach/water solution penetrates into the wood and supplies moisture for continued mold growth while the chlorine simply stays on the surface.

You can try the O. acid or try Tim-bor a commercially available borate based wood preserver. It will have better penetration qualities in liquid form and can also be applied
to kill surface mold as powder. It is a natural pesticide as well as fungicide. I have never used it personally. If you try it let me know how it works.

http://www.custompestcontrol.com/Timbor.html