The purpose of this work is to theoretically and experimentally investigate the applicability of the Tsai–Hill failure criterion and classical laminate theory for predicting the strength and stiffness of 3D-printed polylactide laminate composites with various raster angles in mechanical tests for uniaxial tension and compression. According to the results of tensile and compression tests, the stiffness matrix components of the orthotropic individual lamina and strength were determined. The Poisson’s ratio was determined using the digital image correlation method. It was found that the Tsai–Hill criterion is applicable for predicting the tensile strength and yield strength of laminate polymer composite materials manufactured via fused deposition modeling 3D printing. The calculated values of the elastic moduli for specimens with various raster angles correlate well with the values obtained experimentally. In tensile tests, the error for the laminate with a constant raster angle was 3.3%, for a composite laminate it was 4.4, in compression tests it was 11.9% and 9%, respectively.