There is ample evidence in nature and laboratory experiments that heterogeneous and rough surfaces can exhibit superhydrophobicity. In this study, we investigate the origins of superhydrophobicity using molecular dynamics (MD) simulations of Lennard-Jones liquid droplets on physically patterned surfaces. We consider solid surfaces patterned with both pillars and steps. To facilitate modeling the liquid-solid interaction, we modify Steele's potential to take into account arrays of both steps and pillars characterized by a desired roughness and solid fraction. Using the modified Steele's potential, we achieve considerable computational speed-up in evaluating liquid-solid forces. Simulations of droplet dynamics show that surface roughness and topographic structure have significant effects on the hydrophobic characteristics of the solid surfaces. By varying the surface roughness and/or the solid fraction, we can observe either the Wenzel or the Cassie mode of wetting, as well as mixed-mode wetting. The origins of the wetting modes can be understood in terms of the interplay between the bulk liquid chemical potential and the liquid-solid interfacial tension, as well as the topology of the liquid-solid potential-energy surface induced by surface roughness. The proposed physical mechanisms provide a better understanding of the superhydrophobicity of heterogeneous and rough surfaces.