We assembled novel periodic structures by alternating aligned carbon nanotube (CNT) discs and polycarbonate cylindrical particles for stress damping applications. CNTs are known to present a nonlinear, hysteretic mechanical behavior in compression with the ability to recover from large strains. Furthermore, CNTs are generally more lightweight and thermally stable than currently existing viscoelastic polymers. We test the dynamic response of these CNT/polycarbonate multilayer systems excited by an impact. The response of these systems is found to be highly nonlinear and tunable by variation of the static precompression applied. We studied the wave propagation properties and dissipation in these systems and found that the presence of CNTs improves the energy damping properties of the system when compared with the one observed in polycarbonate particles alone. Such improvement was higher with smaller precompression loads. More detailed studies are required to understand the origin of the observed dynamic behavior.