Mechanical metamaterials are artificial composites with tunable advanced mechanical properties. Particularly, interesting types of mechanical metamaterials are flexible metamaterials, which harness internal rotations and instabilities to exhibit programable deformations. However, to date, such materials have mostly been considered using nearly purely elastic constituents such as neo-Hookean rubbers. Here, we experimentally explore the mechanical snap-through response of metamaterials that are made of constituents that exhibit large viscoelastic relaxation effects, encountered in the vast majority of rubbers, in particular, in 3D printed rubbers. We show that they exhibit a very strong sensitivity to the loading rate. In particular, the mechanical instability is strongly affected beyond a certain loading rate. We rationalize our findings with a compliant mechanism model augmented with viscoelastic interactions, which qualitatively captures well the reported behavior, suggesting that the sensitivity to the loading rate stems from the nonlinear and inhomogeneous deformation rate, provoked by internal rotations. Our findings bring a novel understanding of metamaterials in the dynamical regime and open up avenues for the use of metamaterials for dynamical shape-changing as well as vibration and impact damping applications.