Advances in microfabrication enable the tailoring of surfaces to achieve optimal sorting, mixing, and focusing of complex particulate suspensions in microfluidic devices. Corrugated surfaces have proved to be a powerful tool to manipulate particle motion for a variety of applications, yet the fundamental physical mechanism underlying the hydrodynamic coupling of the suspended particles and surface topography has remained elusive. Here, we study the hydrodynamic interactions between sedimenting spherical particles and nearby corrugated surfaces, whose corrugations are tilted with respect to gravity. Our experiments show three-dimensional, helical particle trajectories with an overall drift along the corrugations, which agree quantitatively with our analytical perturbation theory. The theoretical predictions reveal that the interaction of the disturbance flows, induced by the particle motion, with the corrugations generates locally a transverse anisotropy of the pressure field, which explains the helical dynamics and particle drift. We demonstrate that this dynamical behavior is generic for various surface shapes, including rectangular, sinusoidal, and triangular corrugations, and we identify surface characteristics that produce an optimal particle drift. Our findings reveal a universal feature inherent to particle transport near patterned surfaces and provide fundamental insights for future microfluidic applications that aim to enhance the focusing or sorting of particulate suspensions.