This paper studies the path design problem for cellular-connected unmanned aerial vehicle (UAV), which aims to minimize its mission completion time while maintaining good connectivity with the cellular network. We first argue that the conventional path design approach via formulating and solving optimization problems faces several practical challenges, and then propose a new reinforcement learning-based UAV path design algorithm by applying \emph{temporal-difference} method to directly learn the \emph{state-value function} of the corresponding Markov Decision Process. The proposed algorithm is further extended by using linear function approximation with tile coding to deal with large state space. The proposed algorithms only require the raw measured or simulation-generated signal strength as the input and are suitable for both online and offline implementations. Numerical results show that the proposed path designs can successfully avoid the coverage holes of cellular networks even in the complex urban environment.