We present ALMA, NOEMA, and IRAM-30 m/EMIR observations of the high-density tracer molecules HCN, HCO + , and HNC in three of the brightest lensed dusty star-forming galaxies at z ≃ 3–3.5, part of the Planck ’s Dusty Gravitationally Enhanced subMillimetre Sources (GEMS), with the aim of probing the gas reservoirs closely associated with their exceptional levels of star formation. We obtained robust detections of ten emission lines between J up = 4 and 6, as well as several additional upper flux limits. In PLCK_G244.8+54.9, the brightest source at z = 3.0, the HNC(5–4) line emission at 0.1″ resolution, together with other spatially-integrated line profiles, suggests comparable distributions of dense and more diffuse gas reservoirs, at least over the most strongly magnified regions. This rules out any major effect from differential lensing. This line is blended with CN(4–3) and in this source, we measure a HNC(5–4)/CN(4–3) flux ratio of 1.76 ±0. 86. Dense-gas line profiles generally match those of mid- J CO lines, except in PLCK_G145.2+50.8, which also has dense-gas line fluxes that are relatively lower, perhaps due to fewer dense cores and more segregated dense and diffuse gas phases in this source. The HCO + /HCN ≳ 1 and HNC/HCN ∼ 1 line ratios in our sample are similar to those of nearby ultraluminous infrared galaxies (ULIRGs) and consistent with photon-dominated regions without any indication of important mechanical heating or active galactic nuclei feedback. We characterize the dense-gas excitation in PLCK_G244.8+54.9 using radiative transfer models assuming pure collisional excitation and find that mid- J HCN, HCO + , and HNC lines arise from a high-density phase with an H 2 density of n ∼ 10 5 –10 6 cm −3 , although important degeneracies hinder a determination of the exact conditions. The three GEMS are consistent with extrapolations of dense-gas star-formation laws derived in the nearby Universe, adding further evidence that the extreme star-formation rates observed in the most active galaxies at z ∼ 3 are a consequence of their important dense-gas contents. The dense-gas-mass fractions traced by HCN/[CI] and HCO + /[CI] line ratios are elevated, but not exceptional as compared to other lensed dusty star-forming galaxies at z > 2, and they fall near the upper envelope of local ULIRGs. Despite the higher overall gas fractions and local gas-mass surface densities observed at high redshift, the dense-gas budget of rapidly star-forming galaxies seems to have evolved little between z ∼ 3 and z ∼ 0. Our results favor constant dense-gas depletion times in these populations, which is in agreement with theoretical models of star formation.