It is relevant for various applications to seek a material that could be used for the active control of wave propagation. In this study, we explore tunablemetamaterials based on foldable origami sheets where the size of the bandgap can be tuned by the rigid folding along predefined crease lines. A combination of out-of-plane components and a crease results in the coupling and separation of bands in the band structure and eventually in a (modal) bandgap. The location, size, and vibration isolation performance are optimized and analyzed for a set of geometrical and material parameters, resulting in a potential design of aMiura-Ori structure. The methods used in this paper could be used for any kind of origami structure, which may lead to various new applications in the field of vibrations and acoustics.