This paper formulates a new Minimum Constraint Removal (MCR) motion planning problem in which the objective is to remove the fewest geometric constraints necessary to connect a start and goal state with a free path. I present a probabilistic roadmap motion planner for MCR in continuous configuration spaces. The planner operates by constructing increasingly refined roadmaps, and efficiently solves discrete MCR problems on these networks. A number of new theoretical results are given for discrete MCR, including a proof that it is NP-hard by reduction from SET-COVER, and I describe two search algorithms that perform well in practice. The motion planner is proven to produce the optimal MCR with probability approaching 1 as more time is spent, and its convergence rate is improved with various efficient sampling strategies. The planner is demonstrated on three example applications: generating human-interpretable excuses for failure, motion planning under uncertainty, and rearranging movable obstacles.