A four-stage modified Runge-Kutta scheme augmented with the Davis-Yee symmetric total variation diminishing model in a postprocessing stage has been developed to solve the three-dimensional magnetogasdynamic equations. The spatial discretization is performed using finite difference schemes. To be applicable to complex geometries, the system of governing equations and the system eigenstructure have been expressed in a generalized curvilinear coordinate system. The algorithm is validated by solving benchmark problems and is subsequently used in general applications. The shock-capturing properties are demonstrated by simulation of the magnetic shock tube and comparison with solutions computed with other numerical schemes. The implementation of diffusion terms is validated by solving the Hartmann problem, which admits an analytical solution. The performance of the algorithm in multidimensional problems is illustrated in the simulation of ramp flows and convergent channel flows. The method is shown to be accurate, with the ability to capture the physical phenomena arising in magnetogasdynamic flowfields.