The purpose of routing protocols in a computer network is to maximize throughput. Shortest-path routing protocols have the disadvantage of causing bottlenecks due to their single-path routing. That is, the shortest path between a source and a destination may become highly congested even when many other paths have low utilization. In this paper, we propose a routing scheme that randomly balances traffic over the whole network, therefore, it removes bottlenecks and increases network throughput. For each data message to be sent from a source s to a destination d, the proposed routing protocol randomly chooses an intermediate node e, and routes the data message along the shortest path from s to e. Then, it routes the data message via the shortest path from e to d. This increases the effective bandwidth between each pair of nodes. In our simulation results, we show that this proposed routing protocol distributes traffic evenly over the whole network and increases network throughput.