This Chapter studies the wave dynamics of systems obtained by combining lumped masses with tensegrity units to form one-dimensional mechanical metamaterials. The analyzed tensegrity units behave as elastic springs with mechanical response that can be finely adjusted by applying an initial state of self-stress to the system. One-dimensional chains of tensegrity prisms are shown to exhibit a tunable, frequency bandgap response under small amplitude compression loading in the dynamic regime. Both monoatomic and diatomic mass-spring models are studied, by examining various arrangements of two types of tensegrity prisms and lumped masses. The continuum limits of the analyzed discrete models are also derived, with reference to the small wave number regime, and challenging engineering applications of bandgap metamaterials with tensegrity architecture are discussed.