Cylindrical tensegrity modules are self-stabilizing systems whose structural integrity arises from a balance between tensile forces in the cables and compressive forces in the struts. In this study, the nonlinear dynamic behavior of a tensegrity modular structure is investigated, accounting for both geometric and material nonlinearities. First, the nonlinear dynamic response of a single module is characterized, considering material nonlinear behavior and strut buckling. When subjected to vertical compressive loads, the module collapses into an almost planar configuration, drastically reducing the volume enclosed by the structure and storing elastic energy. The influence of topological (diagonal cables and strut connectivity), geometric (bottom and top radius, height, and cable and strut cross sections), and material (elastic modulus and yield stress) parameters, as well as the pre-stress level, on the mechanical response is quantified. Next, the studied module is used to compose more complex assemblies, and their nonlinear dynamic behavior is analyzed and related to that of the single module. A coupling between vertical and twisting deformation modes, resulting in a screw-like motion, is identified. Both the single module and the modular assemblies exhibit a softening response as the magnitude of the excitation force increases, providing insight into the design of such structural systems.