The possible discovery of the ${\ensuremath{\Theta}}^{+}$ pentaquark has motivated a number of studies of its nature using lattice QCD. While all the analyses thus far have focused on spin-$\frac{1}{2}$ states, here we report the results of the first exploratory study in quenched lattice QCD of pentaquarks with spin $\frac{3}{2}$. For the spin-$\frac{3}{2}$ interpolating field we use a product of the standard $N$ and ${K}^{*}$ operators. We do not find any evidence for the standard lattice resonance signature of attraction (i.e., binding at quark masses near the physical regime) in the ${J}^{P}={\frac{3}{2}}^{\ensuremath{-}}$ channel. Some evidence of binding is inferred in the isoscalar ${\frac{3}{2}}^{+}$ channel at several quark masses, in accord with the standard lattice resonance signature. This suggests that this is a good candidate for the further study of pentaquarks on the lattice.