This paper presents a family of serial chain mechanisms with three degrees of freedom (DOF) by concatenating rigid origami modules. This chained mechanism forms a circular arc shape and can continuously extend, bend, and turn. The mechanism keeps three-DOF regardless of the number of connected modules, and the whole motion can be controlled by determining the configuration of one module at the end. We first describe the geometric construction of the mechanism and its implementation as a rigid origami fabricated from a flat sheet. We then analyze the kinematics of the system to illustrate the configuration space and how the shapes change by manipulating the input parameters. We also synthesize the motions by numerically solving inverse kinematics of the system. We also propose novel torus mechanism with two DOF.