Summary: We discuss a method, that uses many of the same techniques as the finite element method itself, to apply standard multigrid algorithms to unstructured finite element problems. We use maximal independent sets (MISs) as a mechanism to automatically coarsen unstructured grids; the inherent flexibility in the selection of MISs allows for the use of heuristics to improve their effectiveness for a multigrid solver. We present parallel algorithms, based on geometric heuristics, to optimize the quality of MISs and the meshes constructed from them, for use in multigrid solvers for three-dimensional unstructured problems. We discuss parallel isssues of our algorithms, multigrid solvers in general, and the parallel finite element application that we have developed to test our solver on challenging problems. We show that our solver and parallel finite element architecture do indeed scale well, with test problems in three-dimensional large deformation elasticity and plasticity, and with 40 million degree of freedom problem on 240 IBM four-way SMP PowerPC nodes.