The influence of non-Gaussian surface roughness on elastic contacts loaded in both normal and tangential directions has been investigated. A numerical solution method based on the multilevel scheme and incorporating the theorem of Ciavarella/Jaeger has been implemented, which allows fast calculation of partial slip loading conditions, including the energy dissipation for a fully reversed tangential loading cycle. The effect of varying roughness rms, skewness, kurtosis, and correlation lengths on contact areas, stiffness values, and energy dissipation is presented, and the significance of these parameters and of the loading method are discussed. It was found that the energy dissipation can be greatly increased by greater surface roughness. Maps showing how the energy dissipation is distributed within the contact are presented, which provide some explanation for this observation and the scatter that may occur for surfaces of nominally similar roughness. The suitability of these parameters for predicting the contact behavior of rough surfaces is also considered.