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We present an updated version of SKIRT, a three-dimensional (3D) Monte Carlo radiative transfer code developed to simulate dusty galaxies. The main novel characteristics of the SKIRT code are the use of a stellar foam to generate random positions, an efficient combination of eternal forced scattering and continuous absorption, and a new library approach that links the radiative transfer code to the DustEM dust emission library. This approach enables a fast, accurate, and self-consistent calculation of the dust emission of arbitrary mixtures of transiently heated dust grains and polycyclic aromatic hydrocarbons, even for full 3D models containing millions of dust cells. We have demonstrated the accuracy of the SKIRT code through a set of simulations based on the edge-on spiral galaxy UGC 4754. The models we ran were gradually refined from a smooth, two-dimensional, local thermal equilibrium (LTE) model to a fully 3D model that includes non-LTE (NLTE) dust emission and a clumpy structure of the dusty interstellar medium. We find that clumpy models absorb UV and optical radiation less efficiently than smooth models with the same amount of dust, and that the dust in clumpy models is on average both cooler and less luminous. Our simulations demonstrate that, given the appropriate use of optimization techniques, it is possible to efficiently and accurately run Monte Carlo radiative transfer simulations of arbitrary 3D structures of several million dust cells, including a full calculation of the NLTE emission by arbitrary dust mixtures.