The theory of tensor function representations constitutes a rational basis for a consistentma them atical modelling of complex mechanical behaviour of anisotropic materials. The so-ca Uedstructural tensors, which characterize the symmetry group of anisotropy of concern, play a key role inobtaining irreducible and coordinate-free representations for anisotropic tensor functions. In thispaper, based on available properties of Kronecker products of orthogonal transformations, a simplemethod of determining the structural tensors with respect to any given symmetry group is developed.As its application, the structural tensors corresponding to the five transverse isotropy groups, all oftheir finite subgroups, and the symmetry group of the 32 crystal classes, which present the most usualand worthwhile anisotropic symmetry groups, are constructed. In particular, we also show that each ofthese anisotropic symmetry groups can be characterized by only one simple structural tensor.