DNA nanotubes can template the growth of nanowires, orient transmembrane proteins for nuclear magnetic resonance determination, and can potentially act as stiff interconnects, tracks for molecular motors and nanoscale drug carriers. Current methods for the construction of DNA nanotubes result in symmetrical and cylindrical assemblies that are entirely double-stranded. Here, we report a modular approach to DNA nanotube synthesis that provides access to geometrically well-defined triangular and square-shaped DNA nanotubes. We also construct the first nanotube assemblies that can exist in double- and single-stranded forms with significantly different stiffness. This approach allows for parameters such as geometry, stiffness, and single- or double-stranded character to be fine-tuned, and could enable the creation of designer nanotubes for a range of applications, including the growth of nanowires of controlled shape, the loading and release of cargo, and the real-time modulation of stiffness and persistence length within DNA interconnects.