A theoretical and experimental investigation of a new class of a tensegrity-based structural damper is presented. The damper is not only capable of attenuating undesirable structural vibrations, as all conventional dampers, but also capable of completely blocking the transmission of vibration over specific frequency bands by virtue of its periodicity. Such dual functionality distinguishes the tensegrity damper over its counterparts of existing structural dampers. Particular emphasis is placed here in presenting the concept and developing the mathematical model of the dynamics of a unit cell the damper. The model is then coupled with a Floquet–Bloch analysis in order to identify the bandgap characteristics of the damper. The predictions of the mathematical model are validated experimentally using a prototype of the damper which is built using 3D printing. Comprehensive material characterization of the damper is performed followed by a detailed extraction of the static and dynamic behavior of the damper in order to validate the theoretical predictions. Close agreement is observed between theory and experiments. The developed theoretical and experimental techniques provide invaluable means for the design of this new class of dampers, particularly for critical structural applications.