In order to satisfy the requirement of high throughput in most unmanned aerial vehicle (UAV) applications, the potential of integrating millimeter wave (mmWave) communications into UAV networks is explored in this paper. A tractable three-dimensional (3D) spatial model is proposed for evaluating the average downlink performance of UAV networks at mmWave bands, where the locations of UAVs and users are randomly distributed with the aid of a Poisson cluster process. Moreover, a typical 3D antenna model with the uniform planar array is deployed at all UAVs to examine the impact of both azimuth and elevation angles. Based on this framework and an ordered user selection scheme, closed-form approximation equations of coverage probabilities are derived. In a noise-limited scenario, an exact expression is provided, which theoretically demonstrates that the coverage performance can be enhanced by increasing the number of antenna elements. Regarding the altitude of UAVs, there exists an optimal height for maximizing the evaluated coverage probability. Numerical results illustrate that non-line-of-sight transmissions caused by obstacles have negligible effects on the considered system.