We present spectra of a sample of Herbig Ae and Be (HAeBe) stars obtained with the Infrared Spectrograph on the Spitzer Space Telescope. All but one of the Herbig stars show emission from polycyclic aromatic hydrocarbons (PAHs) and seven of the spectra show PAH emission, but no silicate emission at 10 microns. The central wavelengths of the 6.2, 7.7--8.2, and 11.3 micron emission features decrease with stellar temperature, indicating that the PAHs are less photo-processed in cooler radiation fields. The apparent low level of photo processing in HAeBe stars, relative to other PAH emission sources, implies that the PAHs are newly exposed to the UV-optical radiation fields from their host stars. HAeBe stars show a variety of PAH emission intensities and ionization fractions, but a narrow range of PAH spectral classifications based on positions of major PAH feature centers. This may indicate that, regardless of their locations relative to the stars, the PAH molecules are altered by the same physical processes in the proto-planetary disks of intermediate-mass stars. Analysis of the mid-IR spectral energy distributions indicates that our sample likely includes both radially flared and more flattened/settled disk systems, but we do not see the expected correlation of overall PAH emission with disk geometry. We suggest that the strength of PAH emission from HAeBe stars may depend not only on the degree of radial flaring, but also on the abundance of PAHs in illuminated regions of the disks and possibly on the vertical structure of the inner disk as well.