We present provably efficient parallel algorithms for sweep scheduling on unstructured meshes. Sweep scheduling is a commonly used technique in Radiation Transport problems, and involves inverting an operator by iteratively sweeping across a mesh. Each sweep involves solving the operator locally at each cell. However, each direction induces a partial order in which this computation can proceed. On a distributed computing system, the goal is to schedule the computation, so that the length of the schedule is minimized. Several heuristics have been proposed for this problem; see and the references therein; but none of the heuristics have worst case performance guarantees. Here we present a simple, almost linear time randomized algorithm which (provably) gives a schedule of length at most O(log^2 n) times the optimal schedule for instances with n cells, when the communication cost is not considered, and a slight variant, which coupled with a much more careful analysis, gives a schedule of (expected) length O(logmlog log logm) times the optimal schedule for m processors. These are the first such provable guarantees for this problem. We also design a priority based list schedule using these ideas, with the same theoretical guarantee, but much better performance in practice. We complement our theoretical results with extensive empirical analysis. The results show that (i) our algorithm performs very well and has significantly better performance guarantee in practice and (ii) the algorithm compares favorably with other natural and efficient parallel algorithms proposed in the literature.