We present high spectral-resolution optical spectra of 49 Herbig Ae/Be stars in a search for the [OI] 6300 angstrom line. The vast majority of the stars in our sample show narrow (FWHM < 100 km s(-1)) emission lines, centered on the stellar radial velocity. In only three sources is the feature much broader (similar to 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 similar to 10 km s(-1). The presence and strength of the [OI] line emission appears to be correlated with the far-infrared energy distribution of each source: stars with a strong excess at 60 mu m have in general stronger [OI] emission than stars with weaker 60 mu m excesses. We interpret these narrow [OI] 6300 angstrom line profiles as arising in the surface layers of the protoplanetary disks surrounding Herbig Ae/Be stars. A simple model for [OI] 6300 angstrom 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 similar to 5 x 10(-7).