Electromagnetic and thermal properties of a non-conventional polymer nanocomposite based on thermoplastic Polylactic acid (PLA, Ingeo™) filled, in different weight percentage, with multi-walled carbon nanotubes (MWCNTs), graphene nanoplatelets (GNPs), as well as a mixture of both fillers (MWCNTs/GNPs), are analyzed. The combination of notable electrical, thermal, and electromagnetic (EM) properties of the carbon fillers, in concentrations above the percolation threshold, together with the good processability of the PLA matrix gives rise to innovative filaments for 3D printing. In particular, the shielding efficiency (SE) in the frequency range 26–37 GHz of samples increases from 0.20 dB of unfilled PLA up to 13.4 dB for composites containing MWCNTs and GNPs, corresponding to 4% and 95% of SE, respectively. The thermal conductivity of the PLA loaded with 12 wt % of GNPs is 263% higher than that of the unfilled polymer, whereas an improvement of about 99% and 190% is detected for the PLA matrix loaded with MWCNTs and both fillers, respectively. The EM and thermal characterization is combined with a morphological investigation allowing us to correlate the dispersion states of the fillers within the polymer matrix with the observed EM and thermal properties. The EM and thermal characteristics exhibited by the nanocomposites make them suitable for packaging applications of electronic devices with electromagnetic interference (EMI) shielding and thermal dissipation features.