The present study investigates the dynamic and cyclic behavior of mixtures of waste materials, i.e., rigid anthropogenic mineral aggregates (RCA) mixed with recycled soft particles (RTW), based on a series of standard resonant column tests and cyclic torsional shear tests. The laboratory tests presented in this article are part of a larger research project that aims to provide useful insights to facilitate the application of RCA–RTW compositions as geotechnical materials. The impacts of various parameters including shear strain, mean effective stress, and, in particular, rubber content on the shear modulus (G), and damping ratio (D), are considered in detail. Rubber content is considered by the percentage of rubber in the mix weight. In general, the results show that as the RTW content increases, the shear modulus decreases while the damping ratio increases. The largest reduction in the G−modulus values occurs for the highest rubberized mix. The observed damping ratio for pure RCA is approx. three times lower versus rubber-reinforced specimens. The compliance of the behavior of the new RCA–RTW mixtures and pure recycled concrete waste tested under dynamic and cyclic loading is demonstrated. The effects of crushing of the RCA material itself during cyclic loading are visible, and dilution of this process due to the addition of rubber. Furthermore, the test data reveal that the values of the G−modulus and D−ratio at small and medium strain levels are considered independent of the time of vibration.