Magnetohydrodynamic (MHD) models are used for a wide range of plasma physics applications. The system of partial differential equations that characterizes these models is nonlinear with strongly coupled fluid and electromagnetic interaction. As a result, the linear systems that arise from discretization and linearization of the nonlinear problem are difficult to solve. In this talk, we consider a multigrid-preconditioned GMRES method as a solver for the discrete linearized MHD equations. We compare three potential relaxation methods for this system, two of which are motivated by well-known relaxation techniques for the incompressible fluids system and extended for MHD, and the other is a new technique that splits the physics into a magnetics-velocity operator and a Navier-Stokes operator. We discuss the parallelization of these techniques and present results for both serial and parallel experiments of an incompressible, steady-state, resistive MHD problem.