This paper focuses on energy efficient transmission in Amplify-and-Forward (AF) relay networks with bidirectional asymmetric traffic. A joint relay selection and power allocation scheme is designed to minimize the energy consumed per bit transmitted for both one-way and two-way relay networks. Through theoretical analysis, close-form solutions for energy efficient power allocation are derived at each relay. Then the direct transmission mode or the relay consuming the least energy is chosen by the sources according to the feedback of each relay. Simulation results present the considerable energy saving ability of the proposed scheme compared with the extended best worse channel relay selection and uniform power allocation (PA) schemes.