This paper presents a new cable-driven underactuated robotic gripper, called TWISTER Hand. It is designed for adaptable grasping of objects in different shapes, weights, sizes, and textures. Each finger of the gripper is made of a compliant and continuum mechanism inspired by an origami design. This design is converted into a Computer Aided Design (CAD) model and 3D-printed using flexible and rigid polymer composite materials. Two CAD modeling methods for this design were compared in terms of structural stiffness and durability in the printed outcomes. For each design, two soft materials were used for preliminary evaluation of the material effect in these properties. The best combination of the model and material was selected to fabricate the three fingers of the robotic gripper. Each finger has a single cable routed along the structure. All three cables are tied and actuated simultaneously using a single servo motor to generate closing and opening motions in the gripper. TWISTER Hand’s adaptable grasping capability was tested using 36 different objects. The robot’s grasping performance under object pose uncertainties was also experimentally tested and analyzed. This compact, fully-integrated gripper can be attached to a robotic arm for various manipulative tasks.