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In this paper, we present newly explored families of rigid-foldable cylinders and the cellular structures constructed from these cylinders; the families include zonogon extrusion cells, bi-directionally flat-foldable cells, and a novel type of cells, i.e., woven cylinder cells. We show the geometry of such structures to demonstrate their validity, their parametric design method, and their kinetic behaviors. These types of structures exhibit continuous rigid-foldability as well as flat-foldability in one or two directions; further, they have different kinetic properties that are potentially applicable for different purposes. The newly proposed woven cylinder cellular structure is a bi-directionally flat-foldable one-DOF rigid-foldable structure and has a distinctive geometric property: structural stiffness against compression in one of three directions.