Direct numerical simulations of heavy particles suspended in a turbulent fluid are performed to study the rate of inter-particle collisions as a function of the turbulence parameters and particle properties. The particle volume fractions are kept small (∼10−4) so that the system is well within the dilute limit. The fluid velocities are updated using a pseudo-spectral algorithm while the particle forces are approximated by Stokes drag. One unique aspect of the present simulations is that the particles have finite volumes (as opposed to point masses) and therefore particle collisions must be accounted for. The collision frequency is monitored over several eddy turnover times. It is found that particles with small Stokes numbers behave similarly to the prediction of Saffman & Turner (1956). On the other hand, particles with very large Stokes numbers have collision frequencies similar to kinetic theory (Abrahamson 1975). For intermediate Stokes numbers, the behaviour is complicated by two effects: (i) particles tend to collect in regions of low vorticity (high strain) due to a centrifugal effect (preferential concentration); (ii) particle pairs are less strongly correlated with each other, resulting in an increase in their relative velocity. Both effects tend to increase collision rates, however the scalings of the two effects are different, leading to the observed complex behaviour. An explanation for the entire range of Stokes numbers can be found by considering the relationship between the collision frequency and two statistical properties of the particle phase: the radial distribution function and the relative velocity probability density function. Statistical analysis of the data, in the context of this relationship, confirms the relationship and provides a quantitative description of how preferential concentration and particle decorrelation ultimately affect the collision frequency.