Fluid flow through isotropic and anisotropic porous medium systems is investigated for a range of Reynolds numbers corresponding to both Darcy and non-Darcy regimes. A non-dimensional formulation is developed for a Forchheimer approximation of the momentum balance, and lattice Boltzmann simulations are used to elucidate the effects of porous medium characteristics on macroscale constitutive relation parameters. The geometric orientation tensor of the solid phase is posited as a morphological measure of leading-order importance for the description of anisotropic flows. Simulation results are presented to confirm this hypothesis, and parameter correlations are developed to predict closure relation coefficients as a function of porous medium porosity, specific interfacial area of the solid phase, and the geometric orientation tensor. The developed correlations provide improved estimates of model coefficients compared to available estimates and extend predictive capabilities to fully determine macroscopic momentum parameters for three-dimensional flows in anisotropic porous media.