3D printing has gained great popularity due to its main feature of manufacturing complex geometries. The building process by adding successive layers generates mechanical properties that depend on the printing parameters, where build orientation is one of the most relevant factors. Due to this, the characterization of the mechanical response of these pieces is a challenging task of practical importance to estimate their lifespan. The aim of this study is to characterize the mechanical behavior and define a 3D constitutive model of polymer materials commonly used in 3D printing manufacturing. Hence, ABS and PLA were used with a low-cost desktop printer with which specimens were manufactured in two orthogonal orientations: flat and upright. Tensile and compression tests were performed to this end, where the Young's modulus, yield, and maximum stresses were determined. In the tensile tests, the samples with vertical (upright) orientation showed lower values in the evaluated mechanical properties than the corresponding to the horizontal (flat) orientation. However, no significant difference caused by the printing orientations was observed in the compression tests. Different values of Young's modulus and maximum strength were found between tensile and compression tests for the same material and orientation. Moreover, in order to describe the observed material response, a linear isotropic bimodular model is proposed. This constitutive model, which is fed with the previously obtained tensile and compression data, is used in the simulation of a four-point bending test where it is found to adequately represent the experimentally measured elastic behavior in the load-deflection curve. Thus, the combination of experiments and a bimodular constitutive model contributes to making better predictions of the mechanical response of structures made with 3D printing.