We present high spectral-resolution optical spectra of 49 Herbig Ae/Be stars in a search for the O i 6300 Å  line. The vast majority of the stars in our sample show narrow (${\it FWHM} < 100$ km s-1) emission lines, centered on the stellar radial velocity. In only three sources is the feature much broader (~400 km s-1), and strongly blueshifted (-200 km s-1) compared to the stellar radial velocity. Some stars in our sample show double-peaked line profiles, with peak-to-peak separations of ~10 km s-1. The presence and strength of the O i line emission appears to be correlated with the far-infrared energy distribution of each source: stars with a strong excess at 60 μm have in general stronger O i emission than stars with weaker 60 μm excesses. We interpret these narrow O i 6300 Å  line profiles as arising in the surface layers of the protoplanetary disks surrounding Herbig Ae/Be stars. A simple model for O i 6300 Å  line emission due to the photodissociation of OH molecules shows that our results are in quantitative agreement with that expected from the emission of a flared disk if the fractional OH abundance is ~$5 \times 10^{-7}$.