We present new experimental constraints on the elastic, spin-dependent WIMP-nucleon cross section using recent data from the XENON100 experiment, operated in the Laboratori Nazionali del Gran Sasso in Italy. An analysis of 224.6 live days×34 kg of exposure acquired during 2011 and 2012 revealed no excess signal due to axial-vector WIMP interactions with 129Xe and 131Xe nuclei. This leads to the most stringent upper limits on WIMP-neutron cross sections for WIMP masses above 6 GeV/c², with a minimum cross section of 3.5×10(-40) cm² at a WIMP mass of 45 GeV/c², at 90% confidence level.