Cable-driven parallel robots (CDPRs) have great prospects for high-speed applications because of their nature of low inertia and good dynamics. Existing high-speed CDPRs mainly adopt redundant cables to keep positive cable tensions. Redundant cables lead to complex and costly structure, and are likely to cause interference. In this study, a non-redundant CDPR for high-speed translational motions is designed with passive springs and parallel cables. First, the configuration of the CDPR is illustrated, and its kinematics and dynamics are studied. Then, the workspace of the CDPR is discussed in detail. The condition of positive cable tensions is proved. The influence of the springs’ layout on the workspace is analyzed. A method for determining the regular cylindrical operation workspace is proposed. Furthermore, the optimal design method for high-speed CDPRs with passive springs is developed. Performance indices for evaluating the force transmission are defined based on the matrix orthogonal degree. The geometric parameters are optimized based on the workspace and force transmission indices. The stiffness coefficient of the spring is determined based on the acceleration and cable tension requirements. Finally, the proposed CDPR and the traditional CDPR with redundant cables are compared through simulation. The results show that the designed CDPR possesses advantages in energy consumption and simple structure compared to CDPR with redundant cables.