Submillimeter galaxies (SMGs) at $z\gtrsim 1$ are luminous in the far-infrared, and have star formation rates, SFR, of hundreds to thousands of solar masses per year. However, it is unclear whether they are true analogs of local ULIRGs or whether the mode of their star formation is more similar to that in local disk galaxies. We target these questions by using Herschel-PACS to examine the conditions in the interstellar medium (ISM) in far-infrared luminous SMGs at $z\sim 1$–4. We present 70–160 μm photometry and spectroscopy of the [O iv]26 μm, [Fe ii]26 μm, [S iii]33 μm, [Si ii]34 μm, [O iii]52 μm, [N iii]57 μm, and [O i]63 μm fine-structure lines and the S(0) and S(1) hydrogen rotational lines in 13 lensed SMGs identified by their brightness in early Herschel data. Most of the 13 targets are not individually spectroscopically detected; we instead focus on stacking these spectra with observations of an additional 32 SMGs from the Herschel archive—representing a complete compilation of PACS spectroscopy of SMGs. We detect [O i]63 μm, [Si ii]34 μm, and [N iii]57 μm at $\geqslant 3\sigma $ in the stacked spectra, determining that the average strengths of these lines relative to the far-IR continuum are $(0.36\pm 0.12)\times {10}^{-3}$, $(0.84\pm 0.17)\times {10}^{-3}$, and $(0.27\pm 0.10)\times {10}^{-3}$, respectively. Using the [O iii]52 μm/[N iii]57 μm emission line ratio, we show that SMGs have average gas-phase metallicities $\gtrsim {Z}_{\odot }$. By using PDR modeling and combining the new spectral measurements with integrated far-infrared fluxes and existing [C ii]158 μm data, we show that SMGs have average gas densities, n, of $\sim {10}^{1-3}\,{\mathrm{cm}}^{-3}$ and FUV field strengths, ${G}_{0}\sim {10}^{2.2-4.5}$ (in Habing units: $1.6\times {10}^{-3}\,\mathrm{erg}\,{\mathrm{cm}}^{-2}\,{{\rm{s}}}^{-1}$), consistent with both local ULIRGs and lower luminosity star-forming galaxies.