Abstract A simple superstatistical Lagrangian stochastic model (Phys. Fluids 15 (2003) L1; Phys. Rev. Lett. 91 (2003) 84503) that accounts explicitly for fluctuations in the rate of dissipation of turbulent kinetic energy has been shown to be in remarkably close agreement with recently acquired data for unconditional Lagrangian acceleration statistics. In this paper, a more elaborate version of the model is shown to predict correctly the observed conditional dependency of Lagrangian acceleration statistics on velocity. The modeling approach is then extended to the simulation of large/heavy-particle accelerations in turbulence. Model predictions for the distribution of accelerations of 450 μm diameter particles with near-neutral buoyancy are shown to be in excellent agreement with experimental data. Tsallis statistics are found to describe accurately model predictions for distributions of heavy-particle velocities and accelerations.