We develop a hybrid computational method to examine the permeation and hindered diffusion through mechanically loaded (anisotropic) polymer networks. We use the bond-bending lattice spring model to capture the micromechanics of random networks of interconnected filaments coupled with the dissipative particle dynamics to explicitly model the viscous fluid and diffusive objects. Our simulations reveal that the transport properties are independent of the network internal organization and are solely function of the network porosity and degree of anisotropy due to a mechanical deformation. Furthermore, our results indicate that the network permeability under load can be estimated based on the network alignment that is characterized by a second order orientation tensor. Our findings have implications for designing drug delivery agents, paper manufacturing, tissue engineering, and understanding the function of biological systems.