This paper deals with the design and 3D motion control of a radio-controlled, multi-link and free-swimming biomimetic robot fish based on an improved kinematic propulsive model. The performance of the robot fish is determined by the fish's both morphological parameters and kinematic parameters. By ichthyologic theories of propulsion, a design framework taking into consideration of both mechatronic constraints in physical realization and feasibility of control methods is presented, where multiple linked robot fish propelled by a flexible posterior body and an oscillating tail fin can be easily developed. The 3D motion control of robot fish is decomposed into speed control, orientation control and submerging/ascending control. The speed of the swimming fish can be adjusted by changing oscillating frequency, oscillating amplitude and the length of oscillatory part, respectively, and its orientation is tuned by different joint's deflections. The up-down motion is realized by a pectoral mechanism. The experimental results on designed prototypes verify that the presented scheme is effective in design and implementation