This paper studies the packet delivery delay performance in three-dimensional mobile ad hoc networks (3D MANETs). Available work mainly focuses on the performance study in two-dimensional MANETs (2D MANETs), which cannot support delay-intensive applications in 3D MANETs. To explore the packet delivery delay performance in 3D MANETs, this paper adopts two-hop relay algorithm with packet replication for packet routing. With such an algorithm, source node can transmit a packet to at most f distinct relay nodes, which then help to forward the packet to its destination node. The algorithm is flexible such that the packet delivery process can be controlled through a proper setting of f. Specially, a general Markov chain theoretical framework is developed to model the packet delivery process under the algorithm in 3D MANETs. Based on the theoretical framework, the closed-form expressions are further derived for mean and relative standard deviation of packet delivery delay. Finally, extensive simulation and numerical results are provided to validate our theoretical models and illustrate the impact of network parameters on packet delivery delay performance in 3D MANETs.