Summary: Several new discrete surface integral methods for solving Maxwell’s equations in the time-domain are presented. These methods, which allow the use of general non-orthogonal mixed-polyhedral unstructured grids, are direct generalizations of the canonical staggered-grid finite difference method. These methods are conservative in that they locally preserve “divergence” or charge. Employing mixed polyhedral cells (hexahedral, tetrahedral, etc.), these methods allow more accurate modeling of non-rectangular structures and objects because the traditional “stair-stepped” boundary approximations associated with the orthogonal grid based finite difference methods can be avoided. Numerical results demonstrating the accuracy of these new methods are presented.