Vibration suppression remains a crucial issue in the design of structures and machines. Recent studies have shown that with the use of metamaterial inspired structures (or metastructures), considerable vibration attenuation can be achieved. Optimization of the internal geometry of metastructures maximizes the suppression performance. Zigzag inserts have been reported to be efficient for vibration attenuation. It has also been reported that the geometric parameters of the inserts affect the vibration suppression performance in a complex manner. In an attempt to find out the most efficient parameters, an optimization study has been conducted on the linear zigzag inserts and is presented here. The research reported in this paper aims at developing an automated method for determining the geometry of zigzag inserts through optimization. This genetic algorithm based optimization process searches for optimal zigzag designs which are properly tuned to suppress vibrations when inserted in a specific host structure (cantilever beam). The inserts adopted in this study consist of a cantilever zigzag structure with a mass attached to its unsupported tip. Numerical simulations are carried out to demonstrate the efficiency of the proposed zigzag optimization approach.