We present new experimental constraints on the WIMP-nucleon spin-dependent elastic cross sections using data from the first science run of ZEPLIN-III, a two-phase xenon experiment searching for galactic dark matter weakly interacting massive particles based at the Boulby mine. Analysis of approximately 450 kg x days fiducial exposure allow us to place a 90%-confidence upper limit on the pure WIMP-neutron cross section of sigma(n)=1.9x10(-2) pb at 55 GeV/c(2) WIMP mass. Recent calculations of the nuclear spin structure based on the Bonn charge-dependent nucleon-nucleon potential were used for the odd-neutron isotopes 129Xe and 131Xe. These indicate that the sensitivity of xenon targets to the spin-dependent WIMP-proton interaction could be much lower than implied by previous calculations, whereas the WIMP-neutron sensitivity is impaired only by a factor of approximately 2.