Flexible mechanical metamaterials have been recently shown to support a rich nonlinear dynamic response. In particular, it has been demonstrated that the behavior of rotating-square architected systems in the continuum limit can be described by nonlinear Klein-Gordon equations. Here, we report on a general class of solutions of these nonlinear Klein-Gordon equations, namely cnoidal waves based on the Jacobi elliptic functions sn, cn and dn. By analyzing theoretically and numerically their validity and stability in the design- and wave-parameter space, we show that these cnoidal wave solutions extend from linear waves (or phonons) to solitons, while covering also a wide family of nonlinear periodic waves. The presented results thus reunite under the same framework different concepts of linear and non-linear waves and offer a fertile ground for extending the range of possible control strategies for nonlinear elastic waves and vibrations.