  We generalize the standard site percolation model on the $d$-dimensional
lattice to a model on random tessellations of $\mathbb R^d$. We prove the
uniqueness of the infinite cluster by adapting the Burton-Keane argument
\cite{burton1989density}, develop two frameworks that imply the non-triviality
of the phase transition and show that large classes of random tessellations fit
into one of these frameworks. Our focus is on a very general approach that goes
well beyond the typical Poisson driven models. The most interesting examples
might be Voronoi tessellations induced by determinantal processes or certain
classes of Gibbs processes introduced in \cite{schreiber2013}. In a second
paper we will investigate first passage percolation on random tessellations.
