The origin of the high star formation rates (SFR) observed in high-redshift dusty star-forming galaxies is still unknown. Large fractions of dense molecular gas might provide part of the explanation, but there are few observational constraints on the amount of dense gas in high-redshift systems dominated by star formation. In this paper, we present the results of our Atacama large millimeter array (ALMA) program targeting dense-gas tracers (HCN(5-4), HCO ⁺ (5-4), and HNC(5-4)) in five strongly lensed galaxies from the South Pole Telescope (SPT) submillimeter galaxy sample. We detected two of these lines (S/N > 5) in SPT-125-47 at z = 2.51 and tentatively detected all three (S/N ∼ 3) in SPT0551-50 at z = 3.16. Since a significant fraction of our target lines is not detected, we developed a statistical method to derive unbiased mean properties of our sample taking into account both detections and non-detections. On average, the HCN(5-4) and HCO ⁺ (5-4) luminosities of our sources are a factor of ∼1.7 fainter than expected, based on the local L′ _HCN(5-4) - L _IR relation, but this offset corresponds to only ∼2σ if we consider sample variance. We find that both the HCO ⁺ /HCN and HNC/HCN flux ratios are compatible with unity. The first ratio is expected for photo-dominated regions (PDRs) while the second is consistent with PDRs or X-ray dominated regions (XDRs) and/or mid-infrared (IR) pumping of HNC. Our sources are at the high end of the local relation between the star formation efficiency, determined using the L _IR /[CI] and L _IR /CO ratios, and the dense-gas fraction, estimated using the HCN/[CI] and HCN/CO ratios. Finally, in SPT0125-47, which has the highest signal-to-noise ratio, we found that the velocity profiles of the lines tracing dense (HCN, HCO ⁺ ) and lower-density (CO, [CI]) molecular gas are similar. In addition to these lines, we obtained one robust and one tentative detection of ¹³ CO(4-3) and found an average I ¹² CO(4-3) _I ¹³ CO(4-3) flux ratio of 26.1 _-3.5 ^+4.5 , indicating a young but not pristine interstellar medium. We argue that the combination of large and slightly enriched gas reservoirs and high dense-gas fractions could explain the prodigious star formation in these systems.