Fused filament fabrication (FFF) is a promising additive manufacturing (AM) technology due to its ability to build thermoplastics parts with advantages in the design and optimization of models with complex geometries, great design flexibility, recyclability and low material waste. This technique has been extensively used for the manufacturing of conceptual prototypes rather than functional components due to the limited mechanical properties of pure thermoplastics parts. In order to improve the mechanical performance of 3D printed parts based on polymeric materials, reinforcements including nanoparticles, short or continuous fibers and other additives have been adopted. The addition of graphene nanoplatelets (GNPs) to plastic and polymers is currently under investigation as a promising method to improve their working conditions due to the good mechanical, electrical and thermal performance exhibited by graphene. Although research shows particularly promising improvement in thermal and electrical conductivities of graphene-based nanocomposites, the aim of this study is to evaluate the effect of graphene nanoplatelet reinforcement on the mechanical properties, dimensional accuracy and surface texture of 3D printed polylactic acid (PLA) structures manufactured by a desktop 3D printer. The effect of build orientation was also analyzed. Scanning Electron Microscope (SEM) images of failure samples were evaluated to determine the effects of process parameters on failure modes. It was observed that PLA-Graphene composite samples showed, in general terms, the best performance in terms of tensile and flexural stress, particularly in the case of upright orientation (about 1.5 and 1.7 times higher than PLA and PLA 3D850 samples, respectively). In addition, PLA-Graphene composite samples showed the highest interlaminar shear strength (about 1.2 times higher than PLA and PLA 3D850 samples). However, the addition of GNPs tended to reduce the impact strength of the PLA-Graphene composite samples (PLA and PLA 3D850 samples exhibited an impact strength about 1.2–1.3 times higher than PLA-Graphene composites). Furthermore, the addition of graphene nanoplatelets did not affect, in general terms, the dimensional accuracy of the PLA-Graphene composite specimens. In addition, PLA-Graphene composite samples showed, in overall terms, the best performance in terms of surface texture, particularly when parts were printed in flat and on-edge orientations. The promising results in the present study prove the feasibility of 3D printed PLA-graphene composites for potential use in different applications such as biomedical engineering.