In order to achieve better statistical Quality-of-Service (QoS) provisioning for cognitive radio networks (CRN), in this paper, we develop a hybrid underlay/overlay transmission mode for CRNs. Specifically, by applying the theory of effective capacity and taking PN's activity statistics into consideration, we first analyze the maximum achievable throughput of the CRN under two dominant transmission modes, namely underlay and overlay, respectively, and provide efficient algorithms to derive optimal transmission strategies for the two modes. Following the analyses, we then propose a hybrid underlay/overlay transmission mode, through which the cognitive users' QoS requirements can be better guaranteed and network throughput can be further improved. Moreover, we analyze the optimal transmission strategies for both underlay and overlay modes under two limiting cases. Analyses indicate that 1) for the loose QoS requirement, optimal transmission strategies for both underlay and overlay modes become the water-filling algorithm; and 2) for the stringent QoS requirement, the cognitive user will transmit with constant rate. Furthermore, the impact of imperfect channel estimations on our proposed transmission mode is discussed. Simulation results are provided to demonstrate the impacts of delay QoS requirements and PN's activity statistics on maximizing the delay-constrained throughput for both underlay and overlay modes and verify the effectiveness of our proposed transmission mode. Moreover, for the overlay mode, we observe that 1) a unique optimal sensing time exists under the given QoS constraint; and 2) the optimal sensing time surprisingly increases as the QoS constraint gets more stringent.