We present ALMA, NOEMA, and IRAM-30m/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 GEMS sample, to probe the gas reservoirs closely associated with their exceptional levels of star formation. We obtain robust detections of ten emission lines between J_up=4 and 6, as well as several additional flux upper 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. The HCO+/HCN > 1 and HNC/HCN ~ 1 line ratios in our sample are similar to those of nearby ULIRGs and consistent with photon-dominated regions without any indication of important mechanical heating or AGN 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 H2 density n~10^5-6 cm^-3, although important degeneracies prevent determining 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 compared to other lensed dusty star-forming galaxies at z>2 and fall near the upper envelope of local ULIRGs. Our results also favor constant dense-gas depletion times in these populations.