Hmmm... let's see...
I can't think of a general color for the Ainur; they were spirits and it's hard to represent them. The Valar and the Maiar, taken separately, are a different thing, though. But let's stick with the races, for now...
The Elves... Well, we could have white for the Vanyar, anyone of the fire colors - yellow, golden, red - for the Noldor, sea-blue for the Teleri who left Middle-earth and grey for those left behind (with the mention that the Nandor are green), all shades of dark for the Avari (but not black).
The Dwarves - either brown (for the earth) or grey (for the stone).
The Hobbits - any warm, earthly shades.
The Dragons - snake-green, bright red and yellow.
The Eagles - blue, green and brown.
The Trolls - stone grey.
The Orcs - indeed, any shades of rotten. Some could be quite black...
The Ents - green (for the leaves) and brown (for the earth).
The Men... This is really difficult... I think the Houses of the Edain could be represented by: grey, brown and black (Beor); white, golden yellow and blue (Hador); brown, green and black (Haleth). Their descendants, the Numenoreans, by sea-blue, black, golden yellow; the Elendili by white, black and grey. The Rohirrim by white, green and golden yellow. The Easterlings by pale shades of yellow; the Haradrim by red and black.
This is how I see it, anyway... *shrugs* Anyway, it's a nice topic; I'm glad you brought it up, Arwen! :)
Evil may yet be good to have been ... and yet remain evil. (Silmarillion)