Hydraulic permeabilities of fiber networks are of interest for many applications and have been studied extensively. There is little work, however, on permeability calculations in three-dimensional random networks. Computational power is now sufficient to calculate permeabilities directly by constructing artificial fiber networks and simulating flow through them. Even with today's high-performance computers, however, such an approach would be infeasible for large simulations. It is therefore necessary to develop a correlation based on fiber volume fraction, radius, and orientation, preferably by incorporating previous studies on isotropic or structured networks. In this work, the direct calculations were performed, using the finite element method, on networks with varying degrees of orientation, and combinations of results for flows parallel and perpendicular to a single fiber or an array thereof, using a volume-averaging theory, were compared to the detailed analysis. The detailed model agreed well with existing analytical solutions for square arrays of fibers up to fiber volume fractions of 46% for parallel flow and 33% for transverse flow. Permeability calculations were then performed for isotropic and oriented fiber networks within the fiber volume fraction range of 0.3%-15%. When drag coefficients for spatially periodic arrays were used, the results of the volume-averaging method agreed well with the direct finite element calculations. On the contrary, the use of drag coefficients for isolated fibers overpredicted the permeability for the volume fraction range that was employed. We concluded that a weighted combination of drag coefficients for spatially periodic arrays of fibers could be used as a good approximation for fiber networks, which further implies that the effect of the fiber volume fraction and orientation on the permeability of fiber networks are more important than the effect of local network structure.