We report the synthesis of cellulose membranes from balsa wood with an exceptionally high responsivity to humidity change by chemical processing and mechanical compression. By varying the ambient humidity, the produced cellulose membranes can provide a variety of predetermined deformations, such as curve, s-like deformation and curl. The high humidity responsivity is originated from a self-maintained moisture gradient induced by an asymmetrical design of membrane surfaces, aided by the hygroscopic swelling of the cellulose. The moisture-driven actuators are then demonstrated as a three-finger gripper that can grab, hold and release objects 40 times the weight of its own. The combination of natural wood and stimuli-responsive behavior open a way to designing smart structures, actuators and soft robots with environmentally friendly, recyclable and biocompatible materials.