The plastic deformation of polycrystals of magnesium alloy AZ31 with microstructures after severe plastic deformation and annealing deformed in tension at a temperature of 0.52 K was studied. It is established that, regardless of the microstructure, the plastic flow is discontinuously exhibiting stress jumps, whose amplitude increases with increase of strain. With increase of grain size due to annealing, the strength of a polycrystalline material decreases, its plasticity increases, and the discontinuous plastic flow is partially suppressed (the jump amplitude is reduced or even disappears). An analysis of the discontinuous flow statistics showed that in ultrafine grained polycrystals the frequency distribution of the stress jump amplitudes is sharper than that of their coarse-grained counterparts. The results of experiments are discussed employing the concept of the avalanche-like dynamics of dislocations being the cause of discontinuous plastic deformation under conditions of low temperatures and high stresses.