Japanese tatami mats are often arranged so that no four mats meet. This local restriction imposes a rich combinatorial structure when applied to monomino-domino coverings of rectilinear grids. We describe a modular, mechanical game board, prototyped with a desktop 3D printer, that enforces this restriction, and transforms tatami pen-and-paper puzzles into game board puzzles. We review some recent mathematical discoveries on tatami coverings and present five new combinatorial games implemented on the game board.