  A $d$-box is the cartesian product of $d$ intervals of $\mathbb{R}$ and a
$d$-box representation of a graph $G$ is a representation of $G$ as the
intersection graph of a set of $d$-boxes in $\mathbb{R}^d$. It was proved by
Thomassen in 1986 that every planar graph has a 3-box representation. In this
paper we prove that every graph embedded in a fixed orientable surface, without
short non-contractible cycles, has a 5-box representation. This directly
implies that there is a function $f$, such that in every graph of genus $g$, a
set of at most $f(g)$ vertices can be removed so that the resulting graph has a
5-box representation. We show that such a function $f$ can be made linear in
$g$. Finally, we prove that for any proper minor-closed class $\mathcal{F}$,
there is a constant $c(\mathcal{F})$ such that every graph of $\mathcal{F}$
without cycles of length less than $c(\mathcal{F})$ has a 3-box representation,
which is best possible.
