Fatigue crack propagation tests are conducted on magnesium alloy, cruciform specimens under biaxial and uniaxial loadings by using the biaxial fatigue tester which was developed by the authors. We investigate the effect of microstructures of specimens on fatigue crack growth resistance under biaxial stresses. From electron-microscope observations of a fatigue fracture and microstructure observations, it becomes clear that the mechanical properties of a magnesium-alloy, AZ31B, are greatly influenced by the diameter of crystal grains. We find that static tensile strength falls by heat-treating in the high temperature region where the diameter of crystal grains of an X-Y plane becomes large. We also find that the crack progress velocity under equal biaxial stresses gets faster by heat-treating in the high temperature region where the diameter of crystal grain of a Z-X plane becomes large.