Certainly! Here's a potential ruleset for the new variant of chess played on a board with alternating black, white, and grey squares:

1. The board is set up in the same way as standard chess, with the exception that the squares alternate between black, white, and grey.
2. All pieces move and capture as in standard chess, with the exception of the bishop. The bishop may move any number of spaces diagonally on black and white squares, but on grey squares it can only move to adjacent squares and cannot capture any pieces on the grey squares.
3. If a player's king is in check, they must make a move to get out of check. If they cannot, the game is over and their opponent wins.
4. A player can also win by checkmate, where their opponent's king is in check and cannot make a move to get out of check on their next turn.
5. If a player cannot make a legal move on their turn, the game is a stalemate and ends in a draw.

These rules are easy to follow and should result in an interesting game, as the addition of grey squares and the modified bishop movement creates new strategic possibilities for players to consider.
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