Vibration energy harvesting seeks to exploit the energy of ambient random vibration for power generation, particularly in small scale devices. Piezoelectric transduction is often used as a conversion mechanism in harvesting and the random excitation is typically modeled as a Brownian stochastic process. However, non-Brownian excitations are of potential interest, particularly in the nonequilibrium regime of harvester dynamics. In this work, we investigate the averaged power output of a generic piezoelectric harvester driven by Brownian as well as (non-Brownian) Levy stable excitations both in the linear and the Duffing regimes. First, a coupled system of stochastic differential equations that model the electromechanical system are presented. Numerical simulation results (based on the Euler- Maruyama scheme) that show the average power output from the system under Brownian and Levy excitations are presented for the cases where the mechanical degree of freedom behaves as a linear as well as a Duffing oscillator. The results demonstrate that Levy excitations result in higher expectation values of harvested power. In particular, increasing the noise intensity leads to significant increase in power output in the Levy case when compared with Brownian excitations.