For the three-dimensional (3D) propagation scenarios of unmanned aerial vehicle (UAV) communication, a multi-cylinder geometry-based stochastic channel model (GSCM) for non-stationary UAV channels is proposed in this paper. The model allows 3D arbitrary trajectories of the UAV as well as the mobile terminal (MT) on ground. On this basis, a corresponding discrete channel model suitable for hardware implementation is developed. Moreover, a birth-and-death algorithm of random cluster is designed to simulate the time-varying channel characteristics, and a new method, i.e., the summation of several linear frequency modulation signals, is applied to reproduce the non-stationary channel fading. Hardware measurement results show that the output statistical properties, i.e., fading amplitude distribution, delay power spectrum density (DPSD) and Doppler power spectrum density(DPSD) agree well with the theoretical ones, and they also clearly demonstrate the time evolving of channel parameters, e.g., the fading, path delay, path power and Doppler frequency, with the position and velocity variations of the UAV.