Flying ad hoc network (FANET) is a new technology, which creates a self-organized wireless network containing unmanned aerial vehicles (UAVs). In FANET, routing protocols deal with important challenges due to limited energy, frequent link failures, high mobility of UAVs, and limited communication range of UAVs. Thus, a suitable path is always essential to transmit data between UAVs. In this paper, a local filtering-based energy-aware routing scheme (LFEAR) is proposed for FANETs. LFEAR improves the template of the route request (RREQ) packet by adding three fields, namely the energy, reliable distance, and movement similarity of the relevant route to create stable and energy-efficient paths. In the routing process, LFEAR presents a local filtering construction technique to avoid the broadcasting storm issue. This filter limits the broadcasting range of RREQs in the network. Accordingly, only UAVs inside this local filtered area can rebroadcast RREQs and other UAVs must eliminate these packets. After ending the route discovery process, the destination begins the route selection phase and extracts information about each discovered route, including the number of hops, route energy, reliable distance, and movement similarity. Then, the destination node calculates a score for each path based on the extracted information, selects the route with the highest score, and sends a route reply (RREP) packet to the source node through this route. Finally, the simulation process of LFEAR is performed using the NS2 simulator, and two simulation scenarios, namely change in network density and change in the speed of UAVs, are defined to evaluate network performance. In the first scenario, LFEAR improves energy consumption, packet delivery rate, network lifespan, and delay by 1.33%, 1.77%, 6.74%, and 1.71%, while its routing overhead is about 16.51% more than EARVRT. In the second scenario, LFEAR optimizes energy consumption and network lifetime by 5.55% and 5.67%, respectively. However, its performance in terms of routing overhead, packet delivery rate, and delay is 23%, 2.29%, and 6.67% weaker than EARVRT, respectively.