Due to its high mobility, flexible deployment, and low cost, unmanned aerial vehicles (UAVs) have attracted wide attention in wireless communication in recent years. However, the delay requirements (e.g., video streaming, online game, etc.) may limit the UAV's mobility. In this paper, we consider a three-dimensional (3D) UAV communication network, where a UAV is employed to fly flexibly in 3D space to serve ground users with delay requirements. To characterize the fundamental tradeoff between throughput and delay, we introduce the minimum required rate for users and aim to maximize the minimum weighted sum of throughput and required rate for each user, via joint optimization of the 3D UAV trajectory as well as communication time and rate allocation. The formulated problem is a non-convex optimization problem, which is generally intractable. By decomposing the formulated problem into two subproblems, we propose an iterative algorithm by block coordinate descent and difference of two convex (D.C.) optimization as well as successive convex approximation (SCA) techniques. Finally, extensive simulation results show that our proposed solution outperforms baseline schemes and unveils the interesting insights and tradeoff between throughput and delay over 3D UAV communication networks.