Most modern infrastructures are coupled with or are dependent on one another to form multilayer networks. Contrary to a single-layer transportation network, multilayer transportation networks not only have micro-interactions between nodes within each layer, they are also characterized by macro-differences, where the transport speeds of layers differ. Based on a multilayer network model, we articulate an efficient traffic-aware routing (ETAR) strategy. An adjustable macro-parameter and an adjustable micro-parameter are introduced to evaluate the relative importance of transport paths within different layers and reflect the traffic awareness mechanism integrating the global static and the local dynamic information, respectively. A packet can adaptively choose its transport layer and routing direction during the transport process. By optimizing the combination of inter-layer macro-parameter and intra-layer micro-parameter, the ETAR strategy can not only reasonably distribute the traffic load from the low-speed layer to the high-speed layer, but also prevent packets from taking a long detour and waiting at hubs for a long time. The simulation results on both synthetic and empirical networks indicate that the ETAR strategy greatly enhances the traffic capacity of multilayer networks, and that the average path length and the average transport time of packets are guaranteed to be reasonable. We also find that increasing the network size of high-speed layer can obviously improve the traffic capacity, but an appropriate mean degree can help to enhance the network traffic throughput.