A Monte Carlo code (artis) for modelling time-dependent three-dimensional spectral synthesis in chemically inhomogeneous models of Type Ia supernova ejecta is presented. Following the propagation of ?-ray photons, emitted by the radioactive decay of the nucleosynthesis products, energy is deposited in the supernova ejecta and the radiative transfer problem is solved self-consistently, enforcing the constraint of energy conservation in the comoving frame. Assuming a photoionization-dominated plasma, the equations of ionization equilibrium are solved together with the thermal balance equation adopting an approximate treatment of excitation. Since we implement a fully general treatment of line formation, there are no free parameters to adjust. Thus, a direct comparison between synthetic spectra and light curves, calculated from hydrodynamic explosion models, and observations is feasible. The code is applied to the well-known W7 explosion model and the results tested against other studies. Finally, the effect of asymmetric ejecta on broad-band light curves and spectra is illustrated using an elliptical toy model.