Dusty star-forming galaxies at high redshift (1 < z < 3) represent the most intense star-forming regions in the universe. Key aspects to these processes are the gas heating and cooling mechanisms, and although it is well known that these galaxies are gas-rich, little is known about the gas excitation conditions. Only a few detailed radiative transfer studies have been carried out owing to a lack of multiple line detections per galaxy. Here we examine these processes in a sample of 24 strongly lensed star-forming galaxies identified by the Planck satellite (LPs) at z similar to 1.1-3.5. We analyze 162 CO rotational transitions (ranging from J(up) = 1 to 12) and 37 atomic carbon fine-structure lines ([C i]) in order to characterize the physical conditions of the gas in the sample of LPs. We simultaneously fit the CO and [C i] lines and the dust continuum emission, using two different non-LTE, radiative transfer models. The first model represents a two-component gas density, while the second assumes a turbulence-driven lognormal gas density distribution. These LPs are among the most gas-rich, IR-luminous galaxies ever observed (mu L-L(IR(8-1000) (mu m)) similar to 10(13-14.6) L-circle dot; = (2.7 +/- 1.2) x 10(12) M-circle dot, with mu(L) similar to 10-30 the average lens magnification factor). Our results suggest that the turbulent interstellar medium present in the LPs can be well characterized by a high turbulent velocity dispersion ( similar to 100 km s(-1)) and ratios of gas kinetic temperature to dust temperature < T-kin/T-d > similar to 2.5, sustained on scales larger than a few kiloparsecs. We speculate that the average surface density of the molecular gas mass and IR luminosity, Sigma(MISM) similar to 10(3-4) M-circle dot pc(-2) and Sigma(LIR) similar to 10(11-12) L-circle dot kpc(-2), arise from both stellar mechanical feedback and a steady momentum injection from the accretion of intergalactic gas.