A new algorithm for performing parallel Sn sweeps on unstructured meshes is developed. 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, nearly linear speedups on up to 126 processors are observed. This is an important and desirable result, since although analyses of structured meshes indicate that parallel sweeps will not scale with normal partitioning approaches, no severe asymptotic degradation in the parallel efficiency is observed with modest (100) levels of parallelism. This result is a fundamental step in the development of efficient parallel Sn methods.