This paper introduces an efficient motion planning method for on-road driving of the autonomous vehicles, which is based on the rapidly exploring random tree (RRT) algorithm. RRT is an incremental sampling-based algorithm and is widely used to solve the planning problem of mobile robots. However, due to the meandering path, the inaccurate terminal state, and the slow exploration, it is often inefficient in many applications such as autonomous vehicles. To address these issues and considering the realistic context of on-road autonomous driving, we propose a fast RRT algorithm that introduces a rule-template set based on the traffic scenes and an aggressive extension strategy of search tree. Both improvements lead to a faster and more accurate RRT toward the goal state compared with the basic RRT algorithm. Meanwhile, a model-based prediction postprocess approach is adopted, by which the generated trajectory can be further smoothed and a feasible control sequence for the vehicle would be obtained. Furthermore, in the environments with dynamic obstacles, an integrated approach of the fast RRT algorithm and the configuration-time space can be used to improve the quality of the planned trajectory and the replanning. A large number of experimental results illustrate that our method is fast and efficient in solving planning queries of on-road autonomous driving and demonstrate its superior performances over previous approaches.