Multicast routing is establishing a tree which is rooted from the source node and contains all the multicast destinations. A delay bounded routing tree is a tree in which the accumulated delay from the source node to any destination along the tree does not exceed a pre-specified bound. This paper presents a distributed routing protocol which constructs delay bounded routing trees for real-time multicast connections. A constructed routing tree has a near 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 number of simulations have been done to show the network cost of the routing trees generated by our method is better than the other major existing algorithms.