Advances in the development of additive manufacturing materials (AM) and the low availability of studies on the impact response of AM specimens are the main reasons for this paper. Therefore, the influence of building orientation (vertical and horizontal) and the angle of the raster (15°/−75°, 30°/−60°, 45°/−45°, and 0°/90°) on the tensile and impact strength of AM specimens was investigated. The polylactic acid (PLA)-PolyMax, Mediflex and acrylonitrile-butadiene-styrene (ABS) filaments were chosen to provide a comprehensive characterization of AM materials with versatile mechanical properties. The experimental results of this study show that the tensile strength and toughness of PolyMax PLA specimens are comparable to ABS specimens, while Mediflex samples are characterized by their higher toughness, but lower impact force needed to break the samples. The Mediflex Charpy fracture surfaces exhibit a ductile character compared to those of brittle ABS and PLA. Furthermore, fracture surface morphology shows the allocation of voids, which helps us to understand differences in mechanical properties, and allows one to properly interpret the results of the geometrical accuracy of AM specimens with various printing settings.