In this study, we show the formation of frequency band structures in origami-based mechanical metamaterials composed of the Triangulated Cylindrical Origami (TCO). Interestingly, the folding behavior of this structure can exhibit both axial and rotational motions under external excitations. Therefore, these two motions can be strongly coupled with each other, which leads to unique dynamic behavior, particularly wave mixing effects. To analyze the folding behavior of the TCO cells, we model their triangular facets into a network of linear springs. We assemble a 1D chain of multiple TCO unit cells stacked vertically in various arrangements, e.g., changing their stacking sequences and/or orientation angles. We study frequency responses of this system to investigate wave mixing effects between axial and rotational motions under dynamic excitations. This dynamic analysis on the multi-cell structure demonstrates the formation of tunable frequency band structures, which can be manipulated by the arrangement of the unit cells and their initial configurations. By taking advantage of their unique dynamic mechanisms, the origami-based mechanical metamaterials have great potential to be used for controlling structural vibrations in an efficient manner.