What I find to be just as fun (assuming all players are near the same level) is to take turns making a move in a single game. There is no communication. It's actually harder than it sounds! You can also do this with multiple games. (My friend and I are going to host a Tandem simul - where we play 25 people and we take turns making the moves.)
There's also a 3 player bug house. The center board will get both sets of other boards captured pieces, and can choose which other board gets their captured piece for each piece.
Truthfully your game sounds a little complicated. There are actually 3 player chess boards - maybe you should buy 2 and content yourself with that?
This is a fun variation my chess club made for 6+players, we called it triple board since normally we played with 6 players. It needs some work, especially for the end game, so I figured I would share it, and see if there were any recommendations
First off all the players are divided into two teams, 3 on each one, and three boards are set up so they are linked, the a file of 1 board adject to h of the other, and white is set up on one side and black on the other. One team is white and one is black.
The turns are white board one, black board one, white board 2..., and on your turn you can only move the white pieces that you control, the ones that started on your board, (much easier to tell using 3 different sets).
all other rules of chess are the same except these changes.
1) when a team is put in check whoever's move it is now must use any of the teams pieces to get out of check.
2) if a move to get out of check puts another king on the same team in check, the move can or cannot be made, but the king that is still in check will be able to be captured on the opposing teams immediate next move.
3)If a team is put into checkmate whoever move it is (whether or not it is their king in checkmate) may move any of their own pieces. Whoever moves next, (the team delivering checkmate) gets to move whatever piece is required to take the king.
4)If you loses your king, you are dead and your pieces become inactive, but can not be captured, and when your turn comes it is skipped.
5) if a check comes and the next move is a skipped turn (a king is gone), then whoever turn is next for that team must make a move, and the skipped turn takes place whenever there is no check.
6) pawns can be promoted to dead kings, allowing that player back into the game.
7) game ends when a team loses all of its kings.
Other thing we added but do not really effect the actual rules were, that teams could not communicate plans except by calling one of a preset number of time-outs, (during which time each team can plan).
This is the game so far, figured I'd share and see if there are any recommendations.