The tradeoff between energy efficiency (EE) and spectral efficiency (SE) in amplify-and-forward relay networks is studied. Three kinds of transmission strategies are considered: 1) noncooperative transmission; 2) relay transmission without a direct link; and 3) relay transmission with a direct link. The tradeoff between EE and SE is formulated as an optimization problem in which the objective is to maximize EE while satisfying the SE requirement. In the noncooperative transmission strategy, the optimization problem is solved by seeking the optimal source transmission power P_s. It is proven that EE is a strictly quasi-concave function of P_s. In the relay transmission with/without a direct link, the optimization problem is solved by jointly seeking the optimal source transmission power P_s and the optimal relay amplification gain β. It is proven that EE is a strictly quasi-concave function of either P_s or β. An optimal but highly complex 1-D searching (ODS) method and a near-optimal but lower complexity alternate optimization (AOP) method are proposed. Simulation results show that the ODS results are consistent with the 2-D exhaustive searching results. The AOP method can achieve an EE close to that obtained by 2-D exhaustive searching. Moreover, the transmission strategies are compared in terms of EE under the same SE constraint.