We used to play a variation of that in college with the jokers also being used (the colored joker was high trump and the black and white one was 2nd highest trump once trump had been assigned). The kitty had 6 cards (to reflect the addition of the jokers) and the trick pattern 9-5-3 (the kitty being the first trick for the dealer).
'Tis a game best played among close friends IMHO since it can be pretty vicious and you need to be able to walk away from it when you are done.
One three-person board game I’ve played a decent amount is End of the Triumvirate, which is a sorta-Risk style game where each player takes the role of one of the members of the First Triumvirate. It’s fun but can run the risk of king-making where attacking one player too aggressively leaves the third player to win easily. There are rules for 2-players but its definitely meant for 3.
OK, that’s a start, but we need rules for how pieces move through that negative-curvature singularity. It’s pretty straightforward for rooks and pawns, and maybe even for kings, but considerably less obvious for bishops (and by extension queens), less yet for knights. Plus, the obvious rule for kings would end up meaning that a king next to the singularity would have at least one move available to him that a queen in the same position wouldn’t, which would lead to all sorts of wrinkles in gameplay (and adjusting the way queens and bishops move to give them corresponding options would also mean that, for instance, bishops can change the color they stand on.
There are also wrinkles that are introduced whenever you have more than two players in chess, regardless of how the board is laid out. For instance, is it still obligatory to escape from check, or is one allowed to hope that the third player will rescue you? Are there any implications to being checked and then un-checked before you have a turn? What about mated and then un-mated? When a player is mated (whatever that means), what becomes of his remaining pieces? Do they stay on the board, cluttering it up, until they’re incidentally captured? If so, is one of the other kings allowed to move into a space theoretically threatened by them? Can the defeated player perhaps even continue to move them, in an attempt to influence which of his opponents wins? Or are they all erased in one swell foop with the fall of their king?