Tensegrity structures have unique features such as low mass to payload ratio, strength, and robustness. Therefore, they present great potential in robotics, aerospace, and civil engineering. The dynamics of tensegrity robots is highly nonlinear and constrained. As a result, their modeling, simulation, state estimation, and control are non-trivial. Strings in tensegrity structures are usually modeled as linear springs. Utilization of nonlinear elastic/damping elements in tensegrities would further enrich their dynamics and endow them with additional properties, such as multiple equilibrium configurations. In this paper, our preliminary work on the dynamics of actuated tensegrities with strings containing nonlinear elastic and/or damping elements is presented. At first, the formulation of tensegrity dynamics with general nonlinear elastic/damping elements is explored. Later dynamics of tensegrities with negative stiffness honeycombs incorporated into strings are considered. Simulations are performed on three tensegrity systems: two-bar, three-bar, and six-bar structures. Results demonstrate that negative stiffness honeycombs result in nonlinear steady-state response to constant external force, reduced force magnitudes in strings and bars, and increased range of motion.