Various theories beyond the Standard Model predict new particles with masses in the sub-eV range with very weak couplings to ordinary matter which can possess spin-dependent couplings to electrons and nucleons. We report null results of a search for possible exotic spin-dependent couplings of the neutron which could be induced by the exchange of light weakly coupled bosons or spin-gravity coupling conducted using a spin-echo neutron spectrometer. We constrain the products gA2 and gAgV of the axial vector coupling of the neutron to the matter of the Earth through the exchange of a weakly coupled vector boson for force ranges between the metre scale and the radius of the Earth. We also constrain the constants in some theories of exotic spin-gravity couplings.