Bio-inspired hand-like grippers actuated by Shape Memory Alloy (SMA) wires represent an emerging new technology with potential applications in many different fields, ranging from industrial assembly processes to biomedical systems. The inherently high energy density makes SMAs a natural choice for compact, lightweight, and silent actuator systems capable of producing a high amount of work, such as hand prostheses or robotic systems in industrial human/machine environments. In this work, a concept for a compact and versatile gripping system is developed, in which SMA wires are implemented as antagonistic muscles actuating an artificial hand with three fingers. In order to combine high gripping force with sufficient actuation speed, the muscle implementation pursues a multi-wire concept with several 0.1 mm diameter NiTi wires connected in parallel, in order to increase the surface-to-volume ratio for accelerated cooling. The paper starts with an illustration of the design concept of an individual 3-phalanx-finger, along with kinematic considerations for optimal placement of SMA wires. Three identical fingers are subsequently fabricated via 3D printing and assembled into a hand-like gripper. The maximum displacement of each finger phalanx is measured, and an average phalanxes dynamic responsiveness is evaluated. SMA self-sensing is documented by experiments relating the wires change in resistance to the finger motion. Several finger force measurements are also performed. The versatility of the gripper is finally documented by displaying a variety of achievable grasping configurations.