Google image for Japanese cloth patterns, that's a start

but initially look for the blue cloth as there is a lot of modern stuff mixed in.
Not very bright colours used at all really unless you were really high up or 'plying your trade'

Any colours including greys (both warm and cold), light blues to dark blues (on the grey side of things), browns of varying shades again, not too bright. Reds could be used, but not too bright (think more terracotta).
As for the patterns themselves, some Japanese cloth can be quite beautiful and intricate but as these are gaming figures simple repeat patterns would probably do. For example a set of three dots in a triangle repeated over the cloth would look good. Also a short line, like a dash, again repeated. Small squares would work.
If you make the pattern colour a lighter shade of the cloth colour it will help pull it all together

cheers
James