Hey, Joshua. A little too bland for you? Well, there's historical precedent in non-moving pieces - see Gary Gifford's Pillars of Medusa. His other pieces worked okay, though.
Could I propose Blander Chess, where the orthogonal move is not allowed? Pawns would not be able to move except to capture. Rooks and knights, not having a diagonal move, would be stationary. And then there's Blandest Chess, where only leapers may move. We could call that 'The Four Knights Game'. Oh, thinking about the possible moves, I have the last two names [Blander and Blandest] the wrong way around.
Is this too modest a variant? Maybe.