We explore hydrodynamic interactions between microswimmers and corrugated, or rough, surfaces, as found often in biological systems and microfluidic devices. Using the Lorentz reciprocal theorem for viscous flows we derive exact expressions for the roughness-induced velocities up to first order in the surface-height fluctuations and provide solutions for the translational and angular velocities valid for arbitrary surface shapes. We apply our theoretical predictions to elucidate the impact of a periodic, wavy surface on the velocities of microswimmers modeled in terms of a superposition of Stokes singularities. Our findings, valid in the framework of a far-field analysis, show that the roughness-induced velocities vary non-monotonically with the wavelength of the surface. For wavelengths comparable to the swimmer-surface distance a pusher can experience a repulsive contribution due to the reflection of flow fields at the edge of a surface cavity, which decreases the overall attraction to the wall.