Analysis of the mechanical properties of engineering materials with micro-structure generally requires modification of the concept of a simple material. One approach is the theory of micropolar materials which introduces an independent rotation of a material element and the resulting stress and strain tensors are generally non-symmetric. In two-dimensional material models these micro-structures are often represented by geometries which exhibit three-fold symmetry in the plane. In this work we investigate the form of the constitutive relations which this three-fold symmetry imposes. We show that three-fold symmetry requires both the stress and couple stress tensors to be isotropic in the plane. We obtain the constitutive relations for an equilateral triangle structure and for the hexagonal or honeycomb structure, both of which exhibit three-fold symmetry in the plane. These results are compared with the results of previous investigations of these two-dimensional material models.