In this study, composites based on a heterophasic polypropylene (PP) copolymer containing different loadings of micro-sized (i.e., talc, calcium carbonate, and silica) and nano-sized (i.e., a nanoclay) fillers were formulated via melt compounding to obtain PP-based materials suitable for Material Extrusion (MEX) additive manufacturing processing. The assessment of the thermal properties and the rheological behavior of the produced materials allowed us to disclose the relationships between the influence of the embedded fillers and the fundamental characteristics of the materials affecting their MEX processability. In particular, composites containing 30 wt% of talc or calcium carbonate and 3 wt% of nanoclay showed the best combination of thermal and rheological properties and were selected for 3D printing processing. The evaluation of the morphology of the filaments and the 3D-printed samples demonstrated that the introduction of different fillers affects their surface quality as well as the adhesion between subsequently deposited layers. Finally, the tensile properties of 3D-printed specimens were assessed; the obtained results showed that modulable mechanical properties can be achieved depending on the type of the embedded filler, opening new perspectives towards the full exploitation of MEX processing in the production of printed parts endowed with desirable characteristics and functionalities.