The ability to design and control DNA nanodevices with programmed conformational changes has established a foundation for molecular-scale robotics with applications in nanomanufacturing, drug delivery, and controlling enzymatic reactions. The most commonly used approach for actuating these devices, DNA binding and strand displacement, allows devices to respond to molecules in solution, but this approach is limited to response times of minutes or greater. Recent advances have enabled electrical and magnetic control of DNA structures with sub-second response times, but these methods utilize external components with additional fabrication requirements. Here, we present a simple and broadly applicable actuation method based on the avidity of many weak base-pairing interactions that respond to changes in local ionic conditions to drive large-scale conformational transitions in devices on sub-second time scales. To demonstrate such ion-mediated actuation, we modified a DNA origami hinge with short, weakly complementary single-stranded DNA overhangs, whose hybridization is sensitive to cation concentrations in solution. We triggered conformational changes with several different types of ions including mono-, di-, and trivalent ions and also illustrated the ability to engineer the actuation response with design parameters such as number and length of DNA overhangs and hinge torsional stiffness. We developed a statistical mechanical model that agrees with experimental data, enabling effective interpretation and future design of ion-induced actuation. Single-molecule Förster resonance energy-transfer measurements revealed that closing and opening transitions occur on the millisecond time scale, and these transitions can be repeated with time resolution on the scale of one second. Our results advance capabilities for rapid control of DNA nanodevices, expand the range of triggering mechanisms, and demonstrate DNA nanomachines with tunable analog responses to the local environment.