We develop a new algorithm for performing parallel S(sub n) sweeps on unstructured meshes. The algorithm uses a low-complexity list ordering heuristic to determine a sweep ordering on any partitioned mesh. For typical problems and with 'normal' mesh partitionings we have observed nearly linear speedups on up to 126 processors. This is an important and desirable result, since although analyses of structured meshes indicate that parallel sweeps will not scale with normal partitioning approaches, we do not observe any severe asymptotic degradation in the parallel efficiency with modest ((le) 100) levels of parallelism. This work is a fundamental step in the development of parallel S(sub n) methods.