We report the effects of polymer size, concentration, and polymer fluid viscoelasticity on the propulsion kinematics of achiral microswimmers. Magnetically driven swimmer's step-out frequency, orientation angle, and propulsion efficiency are shown to be dependent on fluid microstructure, viscosity, and viscoelasticity. Additionally, by exploring the swimming dynamics of two geometrically distinct achiral structures, we observe differences in propulsion efficiencies of swimmers. Results indicate that larger four-bead swimmers are more efficiently propelled in fluids with significant elasticity in contrast to smaller 3-bead swimmers, which are able to use shear thinning behavior for efficient propulsion. Insights gained from these investigations will assist the development of future microswimmer designs and control strategies targeting applications in complex fluids.