Energy use is a crucial design concern in wireless ad hoc networks since wireless terminals are typically battery-operated. The design objectives of energy-aware routing are two folds: Selecting energy-efficient paths and minimizing the protocol overhead incurred for acquiring such paths. To achieve these goals simultaneously, we present the design of several on-demand energy-aware routing protocols. The key idea behind our design is to adaptively select the subset of nodes that are required to involve in a route-searching process in order to acquire a high residual-energy path and/or the degree to which nodes are required to participate in the process of searching for a low-power path in networks wherein nodes have transmission power adjusting capability. Analytical and simulation results are given to demonstrate the high performance of the designed protocols in energy-efficient utilization as well as in reducing the protocol overhead incurred in acquiring energy-efficient routes.