We generalize the existing finite-size criteria for spectral gaps offrustration-free spin systems to D>2 dimensions. We obtain a local gapthreshold of 3/n, independent of D, for nearest-neighbor interactions. The 1/n scaling persists for arbitrary finite-range interactions in Z3. The key observation is that there is more flexibility in Knabe’s combinatorial approach if one employs the operator Cauchy-Schwarz inequality.