The technique of Brownian dynamics is used to model the electrophoretic mobility of spherical and rod-like particles in a three-dimensional cubic gel lattice. In addition to excluded volume interactions between the migrating particle and the gel, direct interactions are also included. The methodology is first applied to spherical particles in the absence of direct interactions and the resulting mobilities are shown to agree with independent studies. The methodology is then applied to rod-like models of short duplex DNA fragments 10-50 base pairs in length. In the absence of direct interactions between gel and DNA, calculated mobilities show a much weaker dependence on gel concentration than observed in experiments of DNA in Tris-acetate buffer and polyacrylamide gels. When an attractive interaction between gel and DNA of approximately -0.3 k(B)T per base pair at contact is included, good agreement between calculated and experimental mobilities is achieved.