Core-shell nanostructures are particularly interesting for the development of dual-property nanofillers for nanocomposites. In this study, advanced materials compatible with the commonly used Fused Deposition Modeling (FDM) 3D printing technique are reported for heat dissipation applications. Core-shell nanowires based on a highly thermally conductive silver core coated with an electrically insulating silica shell are investigated. The heat dissipation performance of polycarbonate nanocomposites is analyzed using a comprehensive set of thermal, electrical, mechanical, and rheological characterizations to determine the optimal silica nanolayer thickness. We demonstrate that these core-shell nanofillers give access to both high thermal conductivity up to 2.08 +/- 0.05 W m$^{-1}$ K$^{-1}$, and electrically insulating behavior (electrical resistivity > 10$^{12}$ Ω cm) at only 3 vol% loading, while retaining very good mechanical strength. The high dispersion and interfacial cohesion of the nanomaterials with the matrix play a key role in achieving these performances. Moreover, thanks to the alignment of the 1D nanofillers during the FDM printing process, the thermal conductivity of the PC nanocomposite reaches an unprecedented value of 3.48 +/- 0.06 W m$^{-1}$ K$^{-1}$ in the printing direction, i.e. a fifteen-fold increase over the thermal conductivity of neat PC.