Structural DNA nanotechnology is a rapidly emerging field that has demonstrated great potential for applications such as single molecule sensing, drug delivery, and templating molecular components. As the applications of DNA nanotechnology expand, a consideration of their mechanical behavior is becoming essential to understand how these structures will respond to physical interactions. This review considers three major avenues of recent progress in this area: (1) measuring and designing mechanical properties of DNA nanostructures, (2) designing complex nanostructures based on imposed mechanical stresses, and (3) designing and controlling structurally dynamic nanostructures. This work has laid the foundation for mechanically active nanomachines that can generate, transmit, and respond to physical cues in molecular systems.