A technique to solve the periodic homogenization problem is described systematically in this work. The method is to solve the cell problems by imposing eigenstrains in terms of a thermal or a piezoelectric strain to the representative volume element (RVE). Homogenized coefficients are then calculated from stress solutions of those cell problems. As a dual approach, an imposed stress field can also be applied to solve the cell problems. Numerical examples of characterization mechanical properties of complicated microstructure materials are examined. The obtained results show good agreements with the published data. Comparisons show that the technique in this study can be effectively used to characterize the mechanical properties of complex microstructured materials.