In this paper, we tackle the Cyclic Bandwidth Sum Problem, consisting in minimizing the sum of the bandwidth of the edges of an input graph computed in a cycle-structured host graph. This problem has been widely studied in the literature due to its multiple real-world applications, such as circuit design, migration of telecommunication networks, or graph drawing, among others. Particularly, we tackle this problem by proposing a multistart procedure whose main components are a new greedy constructive algorithm and an intensification strategy based on the Variable Neighborhood Search metaheuristic. The constructive procedure introduces two different greedy criteria to determine each step of the construction phase, which can be used for other related problems. Additionally, we illustrate how to perform an efficient exploration of the neighborhood structure by using an alternative neighborhood. Our algorithmic proposal is evaluated over a set of 40 instances previously studied in the literature and over a new proposed set of 66 well-known instances introduced in this paper. The obtained results have been satisfactory compared with the ones obtained by the best previous algorithm in the state of the art. The statistical tests performed indicate that the differences between the methods are significant.