The most intensively star-forming galaxies are extremely luminous at far-infrared ( FIR) wavelengths, highly obscured at optical and ultraviolet wavelengths, and lie at z >= 1-3. We present a programme of FIR spectroscopic observations with the SPIRE FTS, as well as photometric observations with PACS, both on board Herschel, towards a sample of 45 gravitationally lensed, dusty starbursts across z similar to 1-3.6. In total, we detected 27 individual lines down to 3 sigma, including nine [C II] 158 mu m lines with confirmed spectroscopic redshifts, five possible [C II] lines consistent with their FIR photometric redshifts, and in some individual sources a few [OIII] 88 mu m, [O III] 52 mu m, [O I] 145 mu m, [O I] 63 mu m, [N II] 122 mu m and OH 119 mu m ( in absorption) lines. To derive the typical physical properties of the gas in the sample, we stack all spectra weighted by their intrinsic luminosity and by their 500 mu m flux densities, with the spectra scaled to a common redshift. In the stacked spectra, we detect emission lines of [CII] 158 mu m, [N II] 122 mu m, [O III] 88 mu m, [O III] 52 mu m, [O I] 63 mu m and the absorption doublet of OH at 119 mu m, at high fidelity. We find that the average electron densities traced by the [NII] and [O III] lines are higher than the average values in local star-forming galaxies and ULIRGs, using the same tracers. From the [NII]/[C II] and [O I]/[C II] ratios, we find that the [CII] emission is likely dominated by the photodominated regions (PDR), instead of by ionized gas or large-scale shocks.