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Current surgical grippers rely on friction grip, where normal loads (i.e. pinch forces) are translated into friction forces. Operating errors with surgical grippers are often force-related, including tissue slipping out of the gripper because of too low pinch forces and tissue damaging due to too high pinch forces. Here, we prototyped a modular surgical gripper with elastomeric soft pads reinforced in the shear direction with a carbon-fiber fabric. The elastomeric component provides low normal stiffness to maximize contact formation without the need of applying high normal loads (i.e. pinch forces), whereas the carbon-fiber fabric offers high shear stiffness to preserve the formed contact under the lateral loads (i.e. shear forces) that occur during tissue lifting. Additionally, we patterned the pads with a sub-surface micropattern, to further reduce the normal stiffness and increase shear stiffness. The body of the prototype gripper, including shaft, joints, and gripper tips, was fabricated in a single step using 3D printing, followed by manual attachment of the soft pads to the gripper. The gripping performance of the newly developed soft gripper on soft tissues was experimentally compared to reference grippers equipped with metal patterned pads. The soft-pad gripper generated similar gripping forces but significantly lower pinch forces than metal-pad grippers. We conclude that grippers with anisotropic-stiffness pads are promising for secure and gentle tissue grip.