The physics of quantized vortex excitations in atomic Bose-Einstein condensates has been extensively studied in recent years. Although simple vortex lines are relatively easy to create, control, and measure in experiments, it is a lot more difficult to do the same for vortex ring structures. Here we suggest and explore a method for generating and controlling superfluid vortex rings, vortex ring lattices, and other three-dimensional vortex structures in toroidally trapped superfluid Bose-Einstein condensates by using the artificial magnetic field produced by an optical nanofiber.