The effect of three-dimensional roughness on pressure-driven flow of water through a microchannel is examined using a finite volume numerical model. In the case of roughness created by the presence of microposts on the channel walls, the pressure drop is found to increase as a function of the height of the microposts. However, in the case where the space between the microposts is free of shear (for example, when air or vapor is trapped between the micro-roughness elements), the pressure drop decreases. This is a model of superhydrophobic surfaces, such as the lotus leaf surface. It is found that there is a critical configuration of the microchannel that can lead to drag reduction even when microposts are present.