This paper introduces a new approach to neuromorphic photonics in which microcavities exhibiting strong exciton-photon interaction may serve as building blocks of optical spiking neurons. The experimental results demonstrate the intrinsic property of exciton-polaritons to resemble the Leaky Integrate-and-Fire spiking mechanism. It is shown that exciton-polariton microcavities when non-resonantly pumped with a pulsed laser exhibit leaky-integration due to relaxation of the excitonic reservoir, threshold-and-fire mechanism due to transition to Bose-Einstein condensate, and resetting due to stimulated emission of photons. These effects, evidenced in photoluminescence characteristics, arise within sub-ns timescales. The presented approach provides means for energy-efficient ultrafast processing of spike-like laser pulses at the level below 1 pJ/spike.