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We present the average rest-frame spectrum of the final catalog of dusty star-forming galaxies (DSFGs) selected from the South Pole Telescope SZ survey (SPT-SZ) and measured with Band 3 of the Atacama Large Millimeter/submillimeter Array (ALMA). This work builds on the previous average rest-frame spectrum, given in Spilker et al. (2014) for the first 22 sources, and is comprised of a total of 78 sources, normalized by their respective apparent dust masses. The spectrum spans $1.9$$<$z$<$$6.9$ and covers rest-frame frequencies of 240$-$800 GHz. Combining this data with low-J CO observations from the Australia Telescope Compact Array (ATCA), we detect multiple bright line features from $^{12}$CO, $[$CI$]$, and H$_2$O, as well as fainter molecular transitions from $^{13}$CO, HCN, HCO$^+$, HNC, CN, H$_2$O$^+$, and CH. We use these detections, along with limits from other molecules, to characterize the typical properties of the interstellar medium (ISM) for these high redshift DSFGs. We are able to divide the large sample into subsets in order to explore how the average spectrum changes with various galaxy properties, such as effective dust temperature. We find that systems with hotter dust temperatures exhibit differences in the bright $^{12}$CO emission lines, and contain either warmer and more excited dense gas tracers, or larger dense gas reservoirs. These observations will serve as a reference point to studies of the ISM in distant luminous DSFGs (L$_{\mathrm{IR}}$$>$$10^{12}$L$_\odot$), and will inform studies of chemical evolution before the peak epoch of star formation at $z=2-3$.