DNA origami nanotechnology is a recently developed self-assembly process for design and fabrication of complex 3D nanostructures using DNA as a functional material. This paper reviews our recent progress in applying DNA origami to design kinematic mechanisms at the nanometer scale. These nanomechanisms, which we call DNA Origami Mechanisms (DOM), are made of relatively stiff bundles of double-stranded DNA (dsDNA), which function as rigid links, connected by highly compliant single-stranded DNA (ssDNA) strands, which function as kinematic joints. The design of kinematic joints including revolute, prismatic, cylindrical, universal and spherical are presented. The steps as well as necessary software or experimental tools for designing DOM with DNA origami links and joints are detailed. To demonstrate the designs, we presented the designs of Bennett four-bar and crank-slider linkages. Finally, a list of technical challenges such as design automation and computational modeling are presented. These challenges could also be opportunities for mechanism and robotics community to apply well developed kinematic theories and computational tools to the design of nanorobots and nanomachines.