It would work, but it would be a complicated game to play. I put together a roughly gradient-coloured version, with places marked for red, white, and black players. What you really have here is one and a half chessboards, six quarters instead of four. Half of a player’s board faces one opponent, and the other half faces their other opponent. Movement rules are 100% normal except for the six central squares, which allow diagonal movement in six directions rather than the usual four (though you could introduce a rule that limits this, maybe forcing you to stay on the same flat plane or something).

There’s one other thing that would cause trouble and need a new rule, though – in traditional chess, king faces king and queen faces queen, with the practical result that one player has his king on his left and the other has his king on his right. With three players, this can’t work, so you’d probably have to state that all players have the king on the left or on the right, whichever works for you.

I’d be interested in printing out the board I made and giving it a try, though I’m not much of a chess player. Would be interesting at least.