In multicast networks, the objective is to send information to multiple users with a single transmission. This allows for savings in bandwidth since data can be shared across network links. An important problem in multicast networks is the determination of an optimum route to be followed by packets in a multicast group. This is known as the multicast routing problem (MRP). A large number of heuristic algorithms have been proposed in recent years to solve the MRP, which is of great interest to network engineers. In this paper, we propose a hybrid metaheuristic approach for the MRP, where a GRASP is used in conjunction with a variable neighborhood search algorithm to find near-optimal solutions to the MRP. Simulation results show that the proposed technique is superior in terms of solution quality.