Multicast routing is to find a tree which is rooted from the source node and contains all the multicast destinations. A delay bounded routing tree is a tree that the accumulated delay from the source node to any destination along the tree shall not exceed a pre-specified bound. This paper presents a distributed heuristic algorithm which generates routing trees with sub-optimal network cost under the delay bound constraint. The proposed algorithm is fully distributed, efficient in terms of the number of messages required, and flexible in multicast membership changes. A large amount of simulations have been done to show the network cost of the routing trees generated by our algorithm is better than the major existing algorithms.