The book Chess Eccentricities by Major George Hope Verney contains descriptions of several forms of Chess for three players played on modified boards of square cells, similar to the Chinese "Game of Three Kingdoms" San Kuo Ch'i.
And, of course, hexagonal chess lends itself very naturally to games for three players. So there are many such games.
But I think that finding three players is much harder than finding two, and the fact that the game basically involves two players choosing to combine against a third in its first phase makes the game unappealing to many.
Some time ago I played checkers for three and enjoyed it. As far I know, there exists a version of chess on a similar board. I was unable to find a working online server for it. Would you try such chess variety if it was available? What do you think about this type of chess game?