This is a cheap way to do this, but it works. First, take a poly plane with 10 subdivisions in width and height and delete every other face so it looks like image 1 down below. Now, duplicate that plane and rotate it 90 degrees. This makes two poly planes directly on top of each other, yet it looks like one single poly plane. (can you guess where this is going yet?) Take one of the planes and apply your white fur to it. Take the other plane and apply your black fur to it. Walla. You have a checker textured fur.

It is also possible to use this same technique on a character, but it takes a lot of time and I'm not sure what the results would be if you tried to bind a skeleton to it and animate it. You could duplicate your character and delete every other face on the original, then do the same with the duplicate, only offset the pattern in which you delete the faces, then apply the furs to each mesh.

Hope this helps.