Depends on what the board is used for.
Racoon: 3/4 inches
Fox: 5/8 inches
Weasel and squirrel: 1/4 inches
Bobcat, coyote and lynx: 5/8 inches
Mink and marten: 3/8 inches
I am pretty sure the latest issue of Fur-Fish-Game has all the sizes you will need. Some people actually use 3/4 inch boards for almost all animals, exept the small ones, such as muskrat, mink, marten, ?fisher?, weasel, squirrel
The reason that my racoon boards are 3/4 inch thick is becasue that is what they recommend in Canada, where coons are regularly bigger.