The plastic deformation of AZ31 magnesium alloy under tension at temperatures of 4.2-295K is studied as a function of its microstructure following squeeze casting (SC) and after severe plastic deformation (SPD) by hot rolling and equal-channel angular pressing. SPD reduces the average grain size and creates a texture that favors basal-plane dislocation glide. It is found that plastic deformation becomes unstable (serrated) at temperatures of 4.2-25K and more stress jerks occur in the SPD polycrystal than in the SC alloy. The temperature dependence of the yield stress of the alloy is typical of thermally activated unpinning of dislocations from short-range barriers. The ratio of the yield stresses for the SPD and SC alloys at a given temperature is explained by hardening owing to a reduction in grain size and softening owing to a favorable texture. As the grain size is reduced, the rate of strain hardening of the alloy falls off, but its ductility (strain to fracture) increases because of the texture. The strain rate sensitivity of the alloy for T≤100K is independent of microstructure and is determined by intersections with forest dislocations. As the temperature is raised over 150-295K the strain rate sensitivity becomes greater owing to activation of dynamic recovery and an enhanced contribution from diffusion processes during plastic deformation of micrograined materials.