Materials exhibit macroscopic properties that are dependent on the underlying components at lower scales. Computational homogenization using the finite element method (FEM) is often used to determine the effective mechanical properties based on the microstructure. However, the use of FEM might suffer from several difficulties such as mesh generation, application of periodic boundary conditions or computations in presence of material interfaces, and further discontinuities. In this paper, we present an alternative approach for computational homogenization of heterogeneous structures based on the scaled boundary finite element method (SBFEM). Based on quadtrees, we are applying a simple meshing strategy to create polygonal elements for arbitrary complex microstructures by using a relatively small number of elements. We show on selected numerical examples that the proposed computational homogenization technique represents a suitable alternative to classical FEM approaches capable of avoiding some of the mentioned difficulties while accurately and effectively calculating the macroscopic mechanical properties. An example of a two-scale semiconcurrent coupling between FEM and SBFEM is presented, illustrating the complementarity of both approaches.