Polymer composites containing carbon nanofillers are extensively developed for electromagnetic shielding applications, where lightweight and flexible materials are required. One example of the microwave absorbers can be thermoplastic fibers fabricated from copolyamide hot melt adhesives and 7 wt% of multi-walled carbon nanotubes, as presented in this paper. A broadband dielectric spectroscopy confirmed that the addition of carbon nanotubes significantly increased microwave electrical properties of the thin (diameter about 100 μm) thermoplastic fibers. Moreover, the dielectric properties are improved for the thicker fibers, and they are almost stable at the frequency range 26–40 GHz and not dependent on the temperature. The variances in the dielectric properties of the fibers are associated with the degree of orientation of carbon nanotubes and the presence of bundles, which were examined using a high-resolution scanning microscope. Analyzing the mechanical properties of the nanocomposite fibers, as an effect of the carbon nanotubes addition, an improvement in the stiffness of the fibers was observed, together with a decrease in the fibers’ elongation and tensile strength.