  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
$z=2-3$.
