  We introduce a percolation model on $\mathbb{Z}^d$, $d \geq 3$, in which the
discrete lines of vertices that are parallel to the coordinate axis are
entirely removed at random and independently of each other. In this way a
vertex belongs to the vacant set $\mathcal{V}$ if and only if none of the $d$
lines to which it belongs, is removed. We show the existence of a phase
transition for $\mathcal{V}$ as the probability of removing the lines is
varied. We also establish that, in the certain region of parameters space where
$\mathcal{V}$ contains an infinite component, the truncated connectivity
function has power-law decay, while inside the region where $\mathcal{V}$ has
no infinite component, there is a transition from exponential to power-law
decay. In the particular case $d=3$ the power-law decay extends through all the
region where $\mathcal{V}$ has an infinite connected component. We also show
that the number of infinite connected components of $\mathcal{V}$ is either
$0$, $1$ or $\infty$.
