Cognitive radio techniques provide potential solutions to the spectrum crisis problem. In cognitive radio networks, a secondary user (SU) opportunistically accesses the authorised spectrum of a primary user (PU) to transmit its data. In order to avoid collisions with the PU, the SU needs to frequently detect the status of the channel before accessing the spectrum. However, frequent detection of channel status consumes a lot of energy, which is unaffordable for mobile SUs. In this paper, we investigate how to preserve the limited energy of SUs in mobile cognitive networks. Besides the sensing and transmitting actions used in traditional spectrum access policies, we introduce a new sleeping action with which mobile SUs can better exploit the trade-off between their consumption and data transmission throughput. We define a utility function to characterise the effect of sleeping actions, in which correct sleeping actions are rewarded for preserving energy, and incorrect sleeping actions are penalised for wasting data transmission opportunities. We prove that, in the meaning of maximising the mean benefit, the utility function exhibits a threshold-based structure. Based on this structure, we design a new spectrum access strategy with which mobile SUs can chose their optimal actions to achieve the best trade-off between energy consumption and throughput. Simulation results show that our spectrum access strategy greatly reduces energy consumption of mobile SUs by up to 85% and, meanwhile, incurs nearly the same PU collision rate and throughput as in state-of-the-art solutions.