Adding shape and interaction anisotropy to a colloidal particle offers exquisitely tunable routes to engineer a rich assortment of complex-architected structures. Inspired by the hierarchical self-assembly concept with block copolymers and DNA liquid crystals and exploiting the unique assembly properties of DNA, we report here the construction and self-assembly of DNA-based soft-patchy anisotropic particles with a high degree of modularity in the system's design. By programmable positioning of thermoresponsive polymeric patches on the backbone of a stiff DNA duplex with linear and star-shaped architecture, we reversibly drive the DNA from a disordered ensemble to a diverse array of long-range ordered multidimensional nanostructures with tunable lattice spacing, ranging from lamellar to bicontinuous double-gyroid and double-diamond cubic morphologies, through the alteration of temperature. Our results demonstrate that the proposed hierarchical self-assembly strategy can be applied to any kind of DNA nanoarchitecture, highlighting the design principles for integration of self-assembly concepts from the physics of liquid crystals, block copolymers, and patchy colloids into the continuously growing interdisciplinary research field of structural DNA nanotechnology.