  We study a generalization of site percolation on a simple cubic lattice,
where not only single sites are removed randomly, but also entire parallel
columns of sites. We show that typical clusters near the percolation transition
are very anisotropic, with different scaling exponents for the sizes parallel
and perpendicular to the columns. Below the critical point there is a Griffiths
phase where cluster size distributions and spanning probabilities in the
direction parallel to the columns have power law tails with continuously
varying non-universal powers. This region is very similar to the Griffiths
phase in subcritical directed percolation with frozen disorder in the preferred
direction, and the proof follows essentially the same arguments as in that
case. But in contrast to directed percolation in disordered media, the number
of active ("growth") sites in a growing cluster at criticality shows a power
law, while the probability of a cluster to continue to grow shows logarithmic
behavior.
