In this work, we address the inverse kinetics problem of motion planning of soft biomimetic actuators driven by three chambers. Soft biomimetic actuators have been applied in many applications owing to their intrinsic softness. Although a mathematical model can be derived to describe the inverse dynamics of this actuator, it is still not accurate to capture the nonlinearity and uncertainty of the material and the system. Besides, such a complex model is time-consuming, so it is not easy to apply in the real-time control unit. Therefore, developing a model-free approach in this area could be a new idea. To overcome these intrinsic problems, we propose a back-propagation (BP) neural network learning the inverse kinetics of the soft biomimetic actuator moving in three-dimensional space. After training with sample data, the BP neural network model can represent the relation between the manipulator tip position and the pressure applied to the chambers. The proposed algorithm is more precise than the analytical model. The results show that a desired terminal position can be achieved with a degree of accuracy of 2.46% relative average error with respect to the total actuator length.