The recent development of the origami technique has revolutionized DNA-based assembly by allowing the synthesis of arbitrarily shaped and sub-micrometer sized two-dimensional (2D) and three-dimensional (3D) architectures with atomic precision. Such structures offer great potential for the development of nanomechanical elements, mediators and sensors, which however requires detailed understanding of their complex mechanical properties. Using magnetic tweezers, we here present direct mechanical measurements on single DNA origami structures and characterize the bending and torsional rigidity of four- and six-helix bundles assembled by this technique. Compared to duplex DNA, we find the bending rigidities to be greatly increased while the torsional rigidities are only moderately augmented. We present a mechanical model explicitly including the crossovers between the individual helices in the origami structure that can describe the experimentally observed behavior. Our results provide an important basis for future applications of 3D DNA origami structures as rigid scaffolds and force transducers as well as noise suppressors in single-molecule mechanical measurements. Beyond that we show how origami structures can be defined and rigidly interfaced to surfaces, which is an important prerequisite to develop structured and three-dimensional surface modifications with the help of DNA templates.