Fused deposition modeling (FDM) is a low-cost additive manufacturing method with moderate tolerances and high design flexibility. Ample studies are being undertaken for modeling the mechanical properties of FDM by using the Finite Element Method (FEM). The process technique of FDM results in anisotropic inner structures that are affected by the chosen manufacturing parameters. Moreover, composite filaments, such as fiber-reinforced polymers, have anisotropy even in filament form before FDM printing. These anisotropic effects are needed to be examined and incorporated for an adequate model. In order to speed up the design stage, we aim to prepare a practical method for simulating the mechanical properties of FDM-printed fiber-reinforced polymer composites. In this work, we computed the homogenized material properties for various fiber lengths, fiber volume percentages, and fiber orientations by asymptotic homogenization at the microscale. Then, mesoscale simulations are carried out through FEM simulations by incorporating the influences of process parameters. In this way, we demonstrate the effect of various micro- and mesostructural features on the homogenized properties step by step.