I'm still figuring out how to do the set too (we will perform in the spring), and we are in a proscenium, so not quite the same space, but here are some of my crazy and not so crazy ideas:
Periaktoi - Can be useful when changing setting rapidly. May not work too well in a thrust though.Lots of movable set pieces on wheels - Make most of the scene changes about re-arranging the furniture. The cast can maybe run through the aisles of the audience while you do this so it feels like they're going to a new room.Have a projection just off stage, or above stage somewhere of the Clue game board (look into copyright). Maybe make an animation of footsteps moving from room to room when your scene shifts.Have an army of run crew to move pieces on/off stage rapidly. No blackouts, part of the fun is the SPEED and madcap energy of the show, which can also be reflected in the way things get moved around.