We present a Spitzer mid-infrared survey of 42 Fanaroff-Riley class II radio galaxies and quasars from the 3CRR catalog at redshift z < 1. All of the quasars and 45% ± 12% of the narrow-line radio galaxies have a mid-IR luminosity of νL_ν(15 μm) > 8 × 10⁴³ ergs s^-1, indicating strong thermal emission from hot dust in the active galactic nucleus. Our results demonstrate the power of Spitzer to unveil dust-obscured quasars. The ratio of "mid-IR luminous" narrow-line radio galaxies to quasars indicates a mean dust covering fraction of 0.56 ± 0.15, assuming relatively isotropic emission. We analyze Spitzer spectra of the 14 mid-IR luminous narrow-line radio galaxies thought to host hidden quasar nuclei. Dust temperatures of 210-660 K are estimated from single-temperature blackbody fits to the low- and high-frequency ends of the mid-IR bump. Most of the mid-IR luminous radio galaxies have a 9.7 μm silicate absorption trough with optical depth <0.2, attributed to dust in a molecular torus. Forbidden emission lines from high-ionization oxygen, neon, and sulfur indicate a source of far-UV photons in the hidden nucleus. However, we find that the other 55% ± 13% of narrow-line FR II radio galaxies are weak at 15 μm, contrary to single-population unification schemes. Most of these galaxies are also weak at 30 μm. Mid-IR weak radio galaxies may constitute a separate population of nonthermal, jet-dominated sources with low accretion power.