A new analytical model for the permeability of anisotropic porous media composed of a periodic in-line arrangement of long rectangular rods, also referred to as 'fibers', was developed. This analytical permeability model was based on an approximation of the microscopic velocity and pressure fields that develop in the pores of the porous medium. The analytical approximation of the velocity field was assessed by an extensive set of numerical simulations of the microscopic velocity and pressure fields for various solidities and aspect ratios of the rectangular rods. The numerical results were obtained by solving the incompressible Navier-Stokes equations, using a volume-penalizing immersed boundary method in which a binary 'masking function' was used to represent the inner geometry of the fluid domain. At the pore scale, laminar flow develops, which is dominated by viscous effects. Therefore, an analytical approximation of the microscopic velocity field based on Poiseuille flow through long slender channels of variable width was proposed. This extended Poiseuille model was compared to the numerical simulations in case the pressure gradient was imposed either 'transverse' or 'longitudinal' to the solid rods that make up the porous medium. The simulated velocity fields compare quite closely with the Poiseuille model for a range of solidities. Based on the extended Poiseuille flow approximation of the velocity field, an analytical model for the effective, anisotropic permeability was developed, which can be used in macroscopic simulations of porous transport. This permeability model was found to describe the permeability for the 'transverse' and 'longitudinal' configurations accurately for viscous, laminar flow (Re less than or similar to 5) at solidities greater than or similar to 0.35. The proposed permeability model was found to be more reliable, for the transverse direction, if the fibers were positioned relative to the flow such that the longest side of the cross section of a fiber was aligned with the main flow direction. (C) 2013 Elsevier Ltd. All rights reserved.