  First-order Fermi acceleration processes at ultrarelativistic shocks are
studied with Monte Carlo simulations. The accelerated particle spectra are
derived by integrating the exact particle trajectories in a turbulent magnetic
field near the shock. ''Realistic'' features of the field structure are
included. We show that the main acceleration process at superluminal shocks is
the particle compression at the shock. Formation of energetic spectral tails is
possible in a limited energy range only for highly perturbed magnetic fields,
with cutoffs occuring at low energies within the resonance energy range
considered. These spectral features result from the anisotropic character of
particle transport in the downstream magnetic field, where field compression
produces effectively 2D perturbations. Because of the downstream field
compression, the acceleration process is inefficient in parallel shocks for
larger turbulence amplitudes, and features observed in oblique shocks are
recovered. For small-amplitude turbulence, wide-energy range particle spectra
are formed and modifications of the process due to the existence of long-wave
perturbations are observed. In both sub- and superluminal shocks, an increase
of \gamma leads to steeper spectra with lower cut-off energies. The spectra
obtained for the ``realistic'' background conditions assumed here do not
converge to the ``universal'' spectral index claimed in the literature. Thus
the role of the first-order Fermi process in astrophysical sources hosting
relativistic shocks requires serious reanalysis.
