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Unmanned Aerial Vehicles (UAVs) are poised to revolutionize communications. Utilizing their flexibility and fast deployment, we can deliver content in congested areas or provide services in areas without infrastructure. In this paper, we consider a UAV that flies over multiple locations and serves as many users as possible within a given time duration. We study the problem of optimal trajectory design, which we formulate as a mixed-integer linear program. For large instances of the problem where the options for trajectories become prohibitively many, we establish a connection to the orienteering problem, and propose a corresponding greedy algorithm. Simulation results show that the proposed algorithm is fast and yields solutions close to the optimal ones. The proposed algorithm can be used for trajectory planning in strategic content caching or tactical field operations.