This paper presents a novel design of a rotary motor-based actuation system for tendon-driven robotic hands. The system features a two-stage, zero backlash, pretensioned pulley reduction that enables high precision control of tendon displacement and force, as well as back-driveability of the motor. While the modular actuation system can be used to actuate any tendon-driven mechanism, the system is designed to meet or exceed the speed and strength of human fingertips. The actuation system is capable of operating up to 10 Hz before there is significant degradation of the system gain. The capabilities of the system are further demonstrated through the control of a Shadow Dexterous Robot Hand index finger at low (1 Hz) and high (4 Hz) frequencies. A single actuation unit can be duplicated for use in a single-acting actuation scheme ("2N-type") of N joints, but was designed specifically for a double-acting actuation scheme ("N-type") with "push-pull" capabilities. In the former case, one motor is used for each rotation direction of a single revolute joint. In the latter case, one motor can control both directions of the joint, thereby reducing hardware needs and control complexity. To address the challenges of tendon slack in the flexible-link transmission to the robot hand, the system includes a lead screw mechanism for setting a tension preload and springs for maintaining tension during the course of use. An integrated uniaxial load cell allows for monitoring of tendon tension and calculation of finger joint torques.