Effective Knudsen diffusion coefficients are presented for fibrous structures consisting of overlapping fibers. The fibers are distributed randomly in d (d=2 or 3) directions with their axes perpendicular to one direction (d=2) or in the three-dimensional space with no preferred orientation (d=3), or they are grouped into d (d=1, 2, or 3) mutually perpendicular bundles of parallel, randomly overlapping fibers. Effective diffusivities are computed using a Monte Carlo simulation scheme to determine the mean square displacement of molecules traveling in the interior of the porous medium for large travel times. Our results show that structures with fibers distributed randomly in d directions have diffusion coefficients identical, within the accuracy of our simulation, to those of d-directional, parallel fiber structures. Effective Knudsen diffusivities are strongly influenced only by the directionality of the fiber structure, with tridirectional or randomly oriented fiber structures presenting lower percolation thresholds (0.04 vs. 0.11) and higher effective diffusivities than bidirectional or random structures with their axes perpendicular to one direction. The tortuosity factor is in general found to decrease with increasing porosity, approaching for each case, as the porosity goes to unity, the corresponding lower bound that is derived using variational principles.