In the present paper we uncover through numerical simulation the velocity and pressure fields inside a model porous medium composed of an infinitely extending array of staggered square rods. These microtransport simulations allow for the prediction of macrotransport parameters that are of value to the volume-averaged description of fluid motion in porous media. We focus on computing the macroscopic apparent permeability and investigate its dependence on the Reynolds number, the porosity, and the flow direction. For the microtransport simulations a volume-penalizing immersed boundary method is presented that facilitates the computation of fluid transport in porous media, accounting for the full geometrical complexity of the porous medium. We represent porous media on uniform Cartesian grids and separate solid from fluid domains using a binary phase-indicator function. The effect of solid bodies on the fluid motion is modeled using a source term in the momentum equation. This source term approximates the no-slip condition at the solid-fluid interface.