Microstructure evolution under hot deformation was investigated in compression of a magnesium alloy AZ31 at 673K (0.73T(m)). Two kinds of samples were machined along the parallel and transverse direction of the extruded rods. New fine grains are evolved at corrugated grain boundaries at low strains and developed rapidly in grain interiors in the medium range of strain, finally leading to a roughly full evolution of equiaxial fine grains. Kink bands are evolved at corrugated grain boundaries and in grain interiors at low strains. The boundary misorientation of kink band increases rapidly with increasing strain. These characteristics of new grain evolution process are not changed by the orientation of the samples, while the flow behaviors clearly depend on it. It is concluded that new grain evolution can be controlled by a deformation-induced continuous reaction resulting in grain fragmentation by kink bands, i.e. continuous dynamic recrystallization (DRX). The latter is discussed comparing with conventional, i.e. discontinuous, DRX.