Advances in fabrication techniques have led to a proliferation of studies on new mechanical metamaterials, particularly on elastic and linear phenomena (for example, their phonon spectrum and acoustic band gaps). More recently, there has been a growing interest in nonlinear wave phenomena in these systems, and particularly how geometric parameters affect the propagation of high-amplitude nonlinear waves. In this paper, we analytically, numerically, and experimentally demonstrate the propagation of cnoidal waves in an elastic architected material. This class of traveling waves constitutes a general family of nonlinear waves, which reduce to phonons and solitons under suitable limits. Although cnoidal waves were first discovered as solutions to the conservation laws for shallow water, they have subsequently appeared in contexts as diverse as ion plasmas and nonlinear optics, but have rarely been explored in elastic solids. We show that geometrically nonlinear deformations in architected soft elastic solids can result in cnoidal waves. Insights from our analysis will be critical to controlling the propagation of stress waves in advanced materials.