  Aims. We present an implementation of an algorithm for 3D time-dependent
Monte Carlo radiative transfer. It allows one to simulate temperature
distributions as well as images and spectral energy distributions of the
scattered light and thermal reemission radiation for variable illuminating and
heating sources embedded in dust distributions, such as circumstellar disks and
dust shells on time scales up to weeks.
  Methods. We extended the publicly available 3D Monte Carlo radiative transfer
code POLARIS with efficient methods for the simulation of temperature
distributions, scattering, and thermal reemission of dust distributions
illuminated by temporally variable radiation sources. The influence of the
chosen temporal step width and the number of photon packages per time step as
key parameters for a given configuration is shown by simulating the temperature
distribution in a spherical envelope around an embedded central star. The
effect of the optical depth on the temperature simulation is discussed for the
spherical envelope as well as for a model of a circumstellar disk with an
embedded star. Finally, we present simulations of an outburst of a star
surrounded by a circumstellar disk.
  Results. The presented algorithm for time-dependent 3D continuum Monte Carlo
radiative transfer is a valuable basis for preparatory studies as well as for
the analysis of continuum observations of the dusty environment around variable
sources, such as accreting young stellar objects. In particular, the combined
study of light echos in the optical and near-infrared wavelength range and the
corresponding time-dependent thermal reemission observables of variable, for
example outbursting sources, becomes possible on all involved spatial scales.
