In this paper we analyze the flow of a Maxwell fluid in a rigid porous medium using the method of volume averaging. We first present the local volume averaged momentum equation which contains Darcy-scale elastic effects and undetermined integrals of the spatial deviations of the pressure and velocity. A closure problem is developed in order to determine the spatial deviations and thus obtain a closed form of the momentum equation that contains a time-dependent permeability tenser. To gain some insight into the effects of elasticity on the dynamics of flow in porous media, the entire problem is transformed to the frequency domain through a temporal Fourier transform. This leads to a dynamic generalization of Darcy＇s law. Analytical results are provided for the case in which the porous medium is modeled as a bundle of capillary tubes, and a scheme is presented to solve the transformed closure problem for a general microstructure.