We present an approach to overcoming challenges in dynamical dexterity for robots through programmably compliant origami mechanisms. Our work leverages a one-parameter family of flat sheet crease patterns that folds into origami bellows, whose axial compliance can be tuned to select desired stiffness. Concentrically arranged cylinder pairs reliably manifest additive stiffness, extending the programmable range by nearly an order of magnitude and achieving bulk axial stiffness spanning 200–1500 N/m using 8 mil thick polyester-coated paper. Accordingly, we design origami energy-storing springs with a stiffness of 1035 N/m each and incorporate them into a three degree-of-freedom (DOF) tendon-driven spatial pointing mechanism that exhibits trajectory tracking accuracy less than 15% rms error within a (2 cm)^3 volume. The origami springs can sustain high power throughput, enabling the robot to achieve asymptotically stable juggling for both highly elastic (1 kg resilient shotput ball) and highly damped (“medicine ball”) collisions in the vertical direction with apex heights approaching 10 cm. The results demonstrate that “soft” robotic mechanisms are able to perform a controlled, dynamically actuated task.