I don't think there's anything in the game that's strictly tied to the colors of the squares. You can still number the ranks and files from the bottom-left corner and e.g. Bishops would start on opposite colors than usual, but the diagonal moves still work the same.
Though, if you set the board up using the mnemonic that the queens are placed on squares matching their color, i.e. white queen on a white square, black queen on a black square, then the setup would be different and the game along with it. But not by much, what you'd end up is this:
That's just the regular starting position flipped. Since all moves are also symmetrical with regard to left and right, it still technically makes no difference other than that now you'd have to number the squares starting from the lower right to match the notation of a regular chess game. (Counting from lower right, the kings here are at e1 and e8 as in a normal game.)
(Castling is the one move which works differently to the left than to the right, but it's defined by the sides where the king/queen start at, and anyway basically reduces down to "king takes two steps and the rook jumps to the square the king passed" which is direction-independent.)