An approach is introduced for performing rigorous numerical simulations of electromagnetic wave scattering from randomly rough, perfectly conducing surfaces. It is based on a surface integral technique, and consists of determining the unknown electric surface current densities from which the electromagnetic field everywhere can be determined. The method is used to study the scattering of a $p$-polarized beam from an {\em anisotropic} Gaussian, randomly rough, perfectly conducting surface. It is demonstrated that the surface anisotropy gives rise to interesting and pronounced signatures in the angular intensity distribution of the scattered light. The origins of these features are discussed.