We present stacked average far-infrared spectra of a sample of 197 dusty star-forming galaxies (DSFGs) at 0.005<z<4 using about 90% of the Herschel Space Observatory SPIRE Fourier Transform Spectrometer (FTS) extragalactic data archive based on 3.5 years of science operations. These spectra explore an observed-frame 447-1568 GHz frequency range, allowing us to observe the main atomic and molecular lines emitted by gas in the interstellar medium. The sample is subdivided into redshift bins, and a subset of the bins are stacked by infrared luminosity as well. These stacked spectra are used to determine the average gas density and radiation field strength in the photodissociation regions (PDRs) of DSFGs. For the low-redshift sample, we present the average spectral line energy distributions of CO and H2O rotational transitions and consider PDR conditions based on observed [C I] 370 and 609 μm, and CO (7-6) lines. For the high-z (0.8<z<4) sample, PDR models suggest a molecular gas distribution in the presence of a radiation field that is at least a factor of 103 larger than the Milky Way and with a neutral gas density of roughly {10}4.5-{10}5.5 cm-3. The corresponding PDR models for the low-z sample suggest a UV radiation field and gas density comparable to those at high-z. Given the challenges in obtaining adequate far-infrared observations, the stacked average spectra we present here will remain the measurements with the highest signal-to-noise ratio for at least a decade and a half until the launch of the next far-infrared facility. Herschel is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA.