<p>Topology optimization is a well-established engineering technique for determining the optimal material distribution for various applications within a defined design space. In this context, the method holds particular utility in cellular materials, aiming to design microstructural configurations that balance demanding performance metrics while minimizing material weight. This communication introduces an innovative methodology synthesizing principles from finite-volume theory, homogenization methods, and density-based topology optimization to design high-performance periodic cellular materials within an efficient Matlab framework. The proposed approach considers two equivalent paths for determining effective material properties based on the fundamental concept of unit cells within periodic materials, serving as an intermediate step in the methodology. These paths include one based on strain energy equivalence and the other employing the classical micromechanical mean-field theory. Subsequently, the optimality criteria algorithm is employed to update the microstructural topologies by considering specific linear combinations of homogenized elastic components, aiming to achieve desired properties such as maximized shear and bulk moduli or negative Poisson’s ratio for auxetic materials while satisfying a prescribed volume fraction constraints. A fundamental contribution of this efficient procedure is to produce microstructural topologies free from checkerboard patterns in the absence of filtering techniques, as demonstrated through numerical investigations.</p>