Mechanical metamaterials are advanced engineering materials that exhibit unusual properties that cannot be found in nature. The elastic properties (i.e., elastic modulus and Poisson's ratio) of mechanical metamaterials can be tuned by changing the geometry of their fundamental unit cells. This allows for the design of metamaterial lattices with targeted quasi-static properties. However, it is not clear how these freedoms contribute to the dynamic properties of mechanical metamaterials. We, therefore, used experimental modal analysis, numerical simulations, and analytical models to study the dynamic response of meta-structures with different values of the Poisson's ratio. We show that Poisson's ratio strongly affects the damping properties of the considered mechanical metamaterials. In particular, we found an inverse relationship between the damping ratio and the absolute value of the Poisson's ratio of the meta-structures. Our results suggest that architected meta-structures similar to those studied could be tailor-made to improve the dissipative performance of mechanical systems. Geometrical design could play an important role in this regard by providing the possibility to tune the various types of quasi-static and dynamic properties of such mechanical metamaterials.