The South Pole Telescope (SPT) has systematically identified 81 high-redshift, strongly gravitationally lensed, dusty star-forming galaxies (DSFGs) in a 2500 square degree cosmological millimeter-wave survey. We present the final spectroscopic redshift survey of this flux-limited (S-870 mu m > 25 mJy) sample, initially selected at 1.4 mm. The redshift survey was conducted with the Atacama Large Millimeter/submillimeter Array across the 3 mm spectral window, targeting carbon monoxide line emission. By combining these measurements with ancillary data, the SPT sample is now spectroscopically complete, with redshifts spanning 1.9 z < 6.9 and a median of of z = 3.9 +/- 0.2. We present the millimeter through far-infrared photometry and spectral energy density fits for all sources, along with their inferred intrinsic properties. Comparing the properties of the SPT sources to the unlensed DSFG population, we demonstrate that the SPT-selected DSFGs represent the most extreme infrared-luminous galaxies, even after accounting for strong gravitational lensing. The SPT sources have a median star formation rate of 2.3(2) x 10(3)M(circle dot) yr(-1) and a median dust mass of 1.4(1) x 10(9) M-circle dot. However, the inferred gas depletion timescales of the SPT sources are comparable to those of unlensed DSFGs, once redshift is taken into account. This SPT sample contains roughly half of the known spectroscopically confirmed DSFGs at z > 5, making this the largest sample of high-redshift DSFGs to date, and enabling the "high-redshift tail" of extremely luminous DSFGs to be measured. Though galaxy formation models struggle to account for the SPT redshift distribution, the larger sample statistics from this complete and well-defined survey will help inform future theoretical efforts.