We propose a four-wave mixing experiment to probe the elementary excitation spectrum of a nonequilibrium Bose-Einstein condensate of exciton-polaritons under nonresonant pumping. Analytical calculations based on mean-field theory show that this method is able to reveal the characteristic negative energy feature of the Bogoliubov dispersion. Numerical simulations including the finite spatial profile of the excitation laser spot and a weak disorder confirm the practical utility of the method for realistic condensates.