Fine-scale scalar mixing in gas-phase planar turbulent jets is studied using measurements of three-component scalar gradient and scalar energy dissipation rate fields. Simultaneous planar Rayleigh scattering and planar laser-induced fluorescence, applied in parallel planes, yield the three-dimensional scalar field measurements. The spatial resolution is sufficient to permit differentiation in all three spatial directions. The data span a range of outer-scale Reynolds numbers from 3290 to 8330. Direct measurement of the thicknesses of scalar dissipation structures (layers) shows that the thicknesses scale with outer-scale Reynolds number as $\hbox{\it Re}_\delta^{-3/4}$, consistent with Kolmogorov/Batchelor scaling. Average layer thicknesses are described by the relation $\lambda_D=14.5\,\delta\, \hbox{\it Re}_\delta^{-3/4}\hbox{\it Sc}^{-1/2}$. There is no evidence here that Taylor scaling ($\lambda_D\propto\delta\, \hbox{\it Re}_\delta^{-1/2}$) plays a significant role in the scalar dissipation process. The present data resolve a range of length scales from the dissipation scales up to nearly the jet full width, and thus can be used in a priori testing of subgrid models for scalar mixing in large-eddy simulations (LES). Comparison of two models for subgrid scalar variance, a scale-similarity model and a gradient-based model, indicates that the scale-similarity model is more accurate at larger LES filter sizes.