In this work, new electrically conductive composite filaments are developed for the fabrication of conductive paths, 3D printed with FDM technology. These composite materials consist of electrically conductive copper powder and a polymer matrix. The influence of three different polymers (ABS, PLA, PS) on the electrical properties of the composites was examined. Electrical measurements of the composite filaments with the increasing copper powder concentrations, allow identifying the percolation threshold for elaborated composites. Results show that the lowest resistivity (0.156 × 10−5 Ωm) was achieved for the ABS/Cu composite at the 84.6 wt% Cu concentration. The obtained resistivity values are much lower than for other conductive composites and nanocomposites filaments reported in the literature. Voltage-current characteristics determined for each composite material showed that composites have Ohmic characteristics in low voltage regime. At high voltage regime, the electrical power dissipated in the composites caused a rapid increase in temperature. It was discovered that a polymer matrix influences the maximum value of the electrical power that can be dissipated in the filament before losing electrical conductivity. Examples of conductive 3D printed structures made from elaborated composites are also presented.