We report experimental limits on new spin-dependent macroscopic forces between neutrons. We measured the nuclear Zeeman frequencies of a 3He/129Xe maser while modulating the nuclear spin polarization of a nearby 3He ensemble in a separate glass cell. We place limits on the coupling strength of neutron spin-spin interactions mediated by light pseudoscalar particles like the axion [g(p)g(p)/(4pihc)] at the 3 x 10(-7) level for interaction ranges longer than about 40 cm. This limit is about 10(-5) the size of the magnetic dipole-dipole interaction between neutrons.