The mechanical behavior of 2-D periodic cellular materials is investigated using a continuum-based modeling approach. Two micromechanical models are developed on the basis of representative unit cell concept in which skeleton of cellular material is modeled as elastic beams. The ANSYS finite element code is used to solve the beam model of skeleton. Elastic moduli of square and triangular networks comprising the microstructure of the cellular material are calculated based on an equivalent continuum model. This is achieved by equating the stored energy in skeleton of a unit cell to the strain energy of the equivalent continuum under a set of prescribed boundary conditions. A proper displacement-controlled (essential) boundary condition generates a uniform strain field in both models which corresponds to calculation of one elastic modulus at a time. Then, effective Young's modulus and Poisson's ratio of continuum are extracted from the calculated elastic moduli. The dependence of effective elastic constants on relative density and thickness to length ratio of the microstructure is investigated. Furthermore, the in-plane behavior of cellular solids in compression is explored with the help of current modeling. The proposed models may contribute to optimal designs of a new class of materials with tailored geometry and material properties which could be useful in a broad range of structural applications.