In this paper, an energy-efficient cross-layer design framework is proposed for cooperative relaying networks, which takes into account the influence of spectrum utilization probability. Specifically, random arrival traffic is considered and an adaptive modulation and coding (AMC) scheme is adopted in the cooperative transmission system to improve the system performance. The average packet dropping rate of the relay-buffer is studied at first. With the packet dropping rate and stationary distribution of the system state, the closed-form expression of the delay is derived. Then the energy efficiency for relay-assisted transmission is investigated, which takes into account the queueing process of the relay and the source. In this context, an energy efficiency optimization problem is formulated to determine the optimum strategy of power and time allocation for the relay-assisted cooperative system. Finally, the energy efficient switching strategy between the relay assisted transmission and the direct transmission is obtained, where packet transmissions have different delay requirements. In addition, energy efficient transmission policy with AMC is obtained. Numerical results demonstrate the effectiveness of the proposed design improving the energy efficiency.