  We present time-domain Monte Carlo simulations of radio emission from cosmic
ray air showers in the scheme of coherent geosynchrotron radiation. Our model
takes into account the important air shower characteristics such as the lateral
and longitudinal particle distributions, the particle track length and energy
distributions, a realistic magnetic field geometry and the shower evolution as
a whole. The Monte Carlo approach allows us to retain the full polarisation
information and to carry out the calculations without the need for any
far-field approximations. We demonstrate the strategies developed to tackle the
computational effort associated with the simulation of a huge number of
particles for a great number of observer bins and illustrate the robustness and
accuracy of these techniques. We predict the emission pattern, the radial and
the spectral dependence of the radiation from a prototypical 10^17 eV vertical
air shower and find good agreement with our analytical results (Huege & Falcke
2003) and the available historical data. Track-length effects in combination
with magnetic field effects surprisingly wash out any significant asymmetry in
the total field strength emission pattern in spite of the magnetic field
geometry. While statistics of total field strengths alone can therefore not
prove the geomagnetic origin, the predicted high degree of polarisation in the
direction perpendicular to the shower and magnetic field axes allows a direct
test of the geomagnetic emission mechanism with polarisation-sensitive
experiments such as LOPES. Our code provides a robust, yet flexible basis for
detailed studies of the dependence of the radio emission on specific shower
parameters and for the inclusion of additional radiation mechanism in the
future.
