Effective Knudsen diffusion coefficients are presented for fibrous structures consisting of parallel, nonoverlapping or partially overlapping fibers. They are computed by means of a Monte Carlo simulation scheme which is employed to determine the mean square displacement of molecules travelling in the interior of the porous medium for large travel times. The results show that structures of paralle, non-overlapping fibers have smaller effective diffusion coefficients parallel to the fibers than structures of parallel, randomly overlapping fibers of the same porosity and fiber radius, but larger in directions perpendicular to the fibers. Partially overlapping fiber structures are found to exhibit behavior intermediate to those of the two extreme cases. Molecular trajectory computations are also used to obtain results for the structural properties of partially overlapping fiber structures (e.g., porosity and internal surface area, accessible porosity and internal surface area, and percolation threshold), which are compared with some results of the literature for the equivalent problem of partially overlapping disks on a plane.