In astrophysical systems, radiation–matter interactions are important in transferring energy and momentum between the radiation field and the surrounding material. This coupling often makes it necessary to consider the role of radiation when modelling the dynamics of astrophysical fluids. During the last few years, there have been rapid developments in the use of Monte Carlo methods for numerical radiative transfer simulations. Here, we present an approach to radiation hydrodynamics that is based on coupling Monte Carlo radiative transfer techniques with finite-volume hydrodynamical methods in an operator-split manner. In particular, we adopt an indivisible packet formalism to discretize the radiation field into an ensemble of Monte Carlo packets and employ volume-based estimators to reconstruct the radiation field characteristics. In this paper the numerical tools of this method are presented and their accuracy is verified in a series of test calculations. Finally, as a practical example, we use our approach to study the influence of the radiation–matter coupling on the homologous expansion phase and the bolometric light curve of Type Ia supernova explosions.