In this paper, we propose an adaptive Medium Access Control (MAC) protocol for full-duplex (FD) cognitive radio networks in which FD secondary users (SUs) perform channel contention followed by concurrent spectrum sensing and transmission, and transmission only with maximum power in two different stages (called the FD sensing and transmission stages, respectively) in each contention and access cycle. The proposed FD cognitive MAC (FDC-MAC) protocol does not require synchronization among SUs and it efficiently utilizes the spectrum and mitigates the self-interference in the FD transceiver. We then develop a mathematical model to analyze the throughput performance of the FDC-MAC protocol where both half-duplex (HD) transmission (HDTx) and FD transmission (FDTx) modes are considered in the transmission stage. Then, we study the FDC-MAC configuration optimization through adaptively controlling the spectrum sensing duration and transmit power level in the FD sensing stage where we prove that there exists optimal sensing time and transmit power to achieve the maximum throughput and we develop an algorithm to configure the proposed FDC-MAC protocol. Extensive numerical results are presented to illustrate the characteristic of the optimal FDC-MAC configuration and the impacts of protocol parameters and the self-interference cancellation quality on the throughput performance. Moreover, we demonstrate the significant throughput gains of the FDC-MAC protocol with respect to existing half-duplex MAC (HD MAC) and single-stage FD MAC protocols.