The proton recoil technique using plastic scintillators is employed to measure fast-neutron spectra. Associated gamma-ray background is eliminated by using various thicknesses of thin scintillators. The method, based on different ranges of electrons and protons in the detector material, is shown to be useful for measuring neutron spectra over an extended energy region. Four scintillator thicknesses are chosen to cover the neutron energy from 1 to 18 MeV, based on practically 100% transmission of Compton-edge electrons. The pulse-height spectra from different detectors are mixed to give a pulse-height spectrum of the “combined” detector. A general purpose unfolding code is developed to unfold the combined pulse-height spectrum using a combined response matrix that results from the mixing of the calculated (Monte Carlo) response functions for the individual detectors. The neutron spectra of different (α,n) sources and of 252Cf spontaneous fission are measured and compared with published data. The technique developed, although used only for a neutron energy up to 14 MeV, can be applied in practice to higher energies.