By employing a modified modal matching approach, we obtain explicit analytical expressions relating frequency to in-plane wave vector for the surface electromagnetic mode confined at the interface between vacuum and a perfect conductor patterned with a two-dimensional square array of square holes. Our complete analytical formalism takes into account both multiple order waveguide modes and diffracted evanescent waves, hence overcoming the a priori assumptions intrinsic to previous descriptions of the dispersion of these surface waves. We validate our derived dispersion relation through comparison with that recently recorded at microwave frequencies using prism coupling. Finally, we show that diffracted evanescent waves play an important role in determining the dispersion, so that the electric field associated with “designer” surface modes is much more weakly confined to the interface than the field associated with surface plasmons on real metal surfaces.