A grid-free variant of the direct simulation Monte Carlo (DSMC) method is proposed, named the isotropic DSMC (I-DSMC) method, that is suitable for simulating dense fluid flows at molecular scales. The I-DSMC algorithm eliminates all grid artifacts from the traditional DSMC algorithm; it is Galilean invariant and microscopically isotropic. The stochastic collision rules in I-DSMC are modified to yield a non-ideal structure factor that gives consistent compressibility, as first proposed by Donev et al (2008 Phys. Rev. Lett. 101 075902). The resulting stochastic hard-sphere dynamics (SHSD) fluid is empirically found to have the same pair correlation function as a deterministic Hamiltonian system of penetrable spheres interacting with a linear core pair potential, well described by the hypernetted chain (HNC) approximation. We apply a stochastic Enskog kinetic theory to the SHSD fluid to obtain estimates for the transport coefficients that are in excellent agreement with particle simulations over a wide range of densities and collision rates. The fluctuating hydrodynamic behavior of the SHSD fluid is verified by comparing its dynamic structure factor against theory based on the Landau–Lifshitz Navier–Stokes equations. We also study the Brownian motion of a nanoparticle suspended in an SHSD fluid and find a long-time power-law tail in its velocity autocorrelation function consistent with hydrodynamic theory and molecular dynamics calculations.