This paper addresses the energy conservation problem in computing systems. The focus is on energy-efficient routing protocols. We formulated and solved a network-wide optimization problem for calculating energy-aware routing for the recommended network configuration. Considering the complexity of the mathematical models of data center networks and the limitations of calculating routing by solving large-scale optimization problems, and methods described in the literature, we propose an alternative solution. We designed and developed several efficient heuristics for equal-cost multipath (ECMP) and Valiant routing that reduce the energy consumption in the computer network interconnecting computing servers. Implementing these heuristics enables the selection of routing paths and relay nodes based on current and predicted internal network load. The utility and efficiency of our methods were verified by simulation. The test cases were carried out on several synthetic network topologies, giving encouraging results. Similar results of using our efficient heuristic algorithm and solving the optimization task confirmed the usability and effectiveness of our solution. Thus, we produced well-justified recommendations for energy-aware computing system design to conclude the paper.