We present a detailed study of the physical properties of the molecular gas in a sample of 18 molecular gas-rich early-type galaxies (ETGs) from the ATLAS3D sample. Our goal is to better understand the star formation processes occurring in those galaxies, starting here with the dense star-forming gas. We use existing integrated 12CO (1-0, 2-1), 13CO (1-0, 2-1), HCN (1-0) and HCO+ (1-0) observations and new 12 CO (3-2) single-dish data. From these, we derive for the first time the average kinetic temperature, H2 volume density and column density of the emitting gas in a significant sample of ETGs, using a non-local thermodynamical equilibrium theoretical model. Since the CO lines trace different physical conditions than of those the HCN and HCO+ lines, the two sets of lines are treated separately. For most of the molecular gas-rich ETGs studied here, the CO transitions can be reproduced with kinetic temperatures of 10-20 K, H2 volume densities of 103-4 cm-3 and CO column densities of 1018-20 cm-2. The physical conditions corresponding to the HCN and HCO+ gas component have large uncertainties and must be considered as indicative only. We also compare for the first time the predicted CO spectral line energy distributions and gas properties of our molecular gas-rich ETGs with those of a sample of nearby well-studied disc galaxies. The gas excitation conditions in 13 of our 18 ETGs appear analogous to those in the centre of theMilky Way, hence the star formation activity driving these conditions is likely of a similar strength and nature. Such results have never been obtained before for ETGs and open a new window to explore further star-formation processes in the Universe. The conclusions drawn should nevertheless be considered carefully, as they are based on a limited number of observations and on a simple model. In the near future, with higher CO transition observations, it should be possible to better identify the various gas components present in ETGs, as well as more precisely determine their associated physical conditions. To achieve these goals, we show here from our theoretical study, that mid-J CO lines [such as the 12CO (6-5) line] are particularly useful.