  Given a tiling $\mathcal{T}$ of the plane by straight edge polygons, which is
invariant by two independent translations, we construct a family of embedded
triply periodic minimal surfaces which desingularizes
$\mathcal{T}\times\mathbb{R}$. For this purpose, inspired by the work of Martin
Traizet, we open the nodes of singular Riemann surfaces to glue together simply
periodic Karcher saddle towers, each placed at a vertex of the tiling in such a
way that its wings go along the corresponding edges of the tiling ending at
that vertex.
