It remains uncertain which continuum and emission line diagnostics best indicate the bolometric powers of active galactic nuclei (AGNs), especially given the attenuation caused by the circumnuclear material and the possible contamination by components related to star formation. Here we use mid-IR spectra along with multiwavelength data to investigate the merit of various diagnostics of AGN radiative power, including the mid-IR [Ne III] λ25.89 μm and [O IV] λ25.89 μm fine-structure lines, the optical [O III] λ5007 forbidden line, and mid-IR 24 μm, 5 GHz radio, and X-ray continuum emission, for complete samples of 46 2Jy radio galaxies (0.05 < z < 0.7) and 17 3CRR FRII radio galaxies (z < 0.1). We find that the mid-IR [O IV] line is the most reliable indicator of AGN power for powerful radio-loud AGNs. By assuming that the [O IV] is emitted isotropically, and comparing the [O III] and 24 μm luminosities of the broad- and narrow-line AGNs in our samples at fixed [O IV] luminosity, we show that the [O III] and 24 μm emission are both mildly attenuated in the narrow-line compared to the broad-line objects by a factor of ≈2. However, despite this attenuation, the [O III] and 24 μm luminosities are better AGN power indicators for our sample than either the 5 GHz radio or the X-ray continuum luminosities. We also detect the mid-IR 9.7 μm silicate feature in the spectra of many objects but not ubiquitously: at least 40% of the sample shows no clear evidence for these features. We conclude that, for the majority of powerful radio galaxies, the mid-IR lines are powered by AGN photoionization.