In this study, the possibility of using a layered silicate-reinforced polylactic acid (PLA) in additive manufacturing applications was investigated. In particular, the aim of this work was to study the influence of printing temperature in the 3D printing process of PLA/clay nanocomposites. For this reason, two PLA grades (4032D and 2003D, D-isomer content 1.5 and 4, respectively) were melt-compounded by a twin screw extruder with a layered silicate (Cloisite 30B) at 4 wt %. Then, PLA and PLA/clay feedstock filaments (diameter 1.75 mm) were produced using a single screw extruder. Dog-bone and prismatic specimens were 3D printed using the FDM technique at three different temperatures, which were progressively increased from melting temperature (185–200–215 °C for PLA 4032D and 165–180–195 °C for PLA 2003D). PLA and PLA/clay specimens were characterized using thermogravimetric analysis (TGA), dynamic mechanical analysis (DMA), differential scanning calorimetry (DSC), and tensile tests. Moreover, the morphology of the 3D printed specimens was investigated using optical microscopy and contact angle measurements. The different polymer matrix and the resulting nanocomposite morphology strongly influenced 3D printed specimen properties. DMA on PLA/clay filaments reported an increase in storage modulus both at ambient temperature and above the glass transition temperature in comparison to neat PLA filaments. Furthermore, the presence of nanoclay increased thermal stability, as demonstrated by TGA, and acted as a nucleating agent, as observed from the DSC measurements. Finally, for 3D printed samples, when increasing printing temperature, a different behavior was observed for the two PLA grades and their nanocomposites. In particular, 3D printed nanocomposite samples exhibited higher elastic modulus than neat PLA specimens, but for PLA 4032D+C30B, elastic modulus increased at increasing printing temperature while for PLA 2003D+C30B slightly decreased. Such different behavior can be explained considering the different polymer macromolecular structure and the different nanocomposite morphology (exfoliated in PLA 4032D matrix and intercalated in PLA 2003D matrix).