A class of exact solutions of the full, viscous magnetohydrodynamic equations are generalized. This class of solutions has the property that the nonlinear terms in the governing equations are not zero identically but as a whole they cancel each other such that the remainder is linear and solvable. The governing equations are formulated using general orthogonal coordinates. Examples in Cartesian coordinates are illustrated. These exact solutions represent a wide variety of flows with probable astrophysical or engineering applications. In each case, the velocity field and the magnetic field are completely coupled.