Multicast networking is poised to play a prominent role in the future deployment of multimedia applications. Multimedia applications are usually resource intensive, have stringent quality of service requirements, and in many cases involve large multicast groups. Multicasting enables these applications to scale to a large number of users without overloading the network and server resources. This paper focuses on the design and analysis of a new heuristic for building low-cost, delay-bounded multicast trees to support multimedia applications. The approach used by the proposed heuristic to "optimize" the multicast tree limits its paths computation to paths which originate from the multicast destinations instead of all pairs of paths, as is the case in other well-known algorithms. Moreover, the path finding procedure, based on Dijkstra's algorithm, resutls in the selection of better paths than those discovered the k-shortest path based procedures. The simulation results show that the proposed heuristic outperforms on average other known heuristics.