The research presented in this paper focuses on a unique multifunctional structural design that not only absorbs vibration at desired frequency bands, but also extracts significant amounts of electrical energy. This is accomplished by first designing an array of low-frequency resonators to be integrated into a larger host structure. This array of resonators can contribute not only to static requirements, e.g., stiffness, strength, mass, etc., of the host structure but the array also functions as a distributed system of passive vibration absorbers. Structures having these distributed vibration absorber systems are known as metastructures. Here, the authors present a unique absorber design referred to as a zigzag beam, which can have a natural frequency an order of magnitude lower than that of a basic cantilever beam of the same scale. It will be shown that the zigzag beams can be designed with an added layer of piezoelectric material, which allows them to harvest significant amounts of electrical power as they suppress vibration of the host structure. This paper includes details of the fully-coupled electromechanical analytical and numerical models for energy harvesting metastructures. Experimental results used to validate the proposed modeling methods will be discussed. Lastly, results of a multi-objective design optimization will be presented and discussed. Results of the optimization study were able to show that allowing only an 82 % increase in the host structure vibration could yield more than a 1500 % increase in total power output. Other results show that the power output (or absorber motion) could be increased 241% without increasing host structure vibrations due to multiple design solutions existing at fixed host structure vibration levels.