A fundamental problem in quality-of-service (QoS) routing is to find a path connecting a source node to a destination node that satisfies K ≥ 2 additive QoS constraints. This multiconstrained path problem (MCP) is known to be NP-complete. In a recent paper, Xue et al. showed that the shortest path with respect to a single auxiliary edge weight (obtained by combining the K edge weights into a single metric) is a K-approximation to MCP, in the sense that the largest ratio of path weight over its corresponding constraint is within a factor of K from minimum. In this paper, we present a simple greedy algorithm and prove that this greedy algorithm is also a K-approximation algorithm to MCP. Extensive computational results show that this greedy algorithm is superior to the previously best known K-approximation algorithm in terms of the quality of the path computed. Our algorithm is as simple as Dijkstra's shortest path algorithm, and is therefore suitable for implementation in Internet protocols.