The material symmetry of the constitutive law of a continuum material is described by the Kronecker powers of the orthogonal tensors which belong to the so-called material symmetry group, a subgroup of the full orthogonal tensor group, of the material. The properties, especially the canonical representations, of Kronecker powers of orthogonal tensors may be applied to deal with material symmetry problems. In this paper, we obtain the basic recurrence formulae in order to determine the canonical representations for finite order Kronecker powers of any given orthogonal tensor; and by usingthe recurrence formulae we derive the canonical representations for first, second, third and fourth order Kronecker powers of any two or three-dimensional orthogonai tensor. Finally, we apply these results to construct the micropolar elasticity matrices for micropolar elastic tensors under the 13 anisotropic mechanics symmetry groups Cn=1, 2....,13 as well as the isotropic symmetry group Co; and we also explain how to find an appropriate orthogonal tensor subgroup which may be regarded as the idealized material symmetry group for a given tensor.