Many microorganisms swim in a highly heterogeneous environment with obstacles such as fibers or polymers. To better understand how this environment affects microorganism swimming, we study propulsion of a cylinder or filament in a fluid with a sparse, stationary network of obstructions modeled by the Brinkman equation. The mathematical analysis of swimming speeds is investigated by studying an infinite-length cylinder propagating lateral or spiral displacement waves. For fixed bending kinematics, we find that swimming speeds are enhanced due to the added resistance from the fibers. In addition, we examine the work and the torque exerted on the cylinder in relation to the resistance. The solutions for the torque, swimming speed, and work of an infinite-length cylinder in a Stokesian fluid are recovered as the resistance is reduced to zero. Finally, we compare the asymptotic solutions with numerical results for the Brinkman flow with regularized forces. The swimming speed of a finite-length filament decreases as its length decreases and planar bending induces an angular velocity that increases linearly with added resistance. The comparisons between the asymptotic analysis and computation give insight on the effect of the length of the filament, the permeability, and the thickness of the cylinder in terms of the overall performance of planar and helical swimmers.