Saddle towers in Heisenberg space

20 pages. V2: addition of a result. V3: minor corrections.We construct most symmetric Saddle towers in Heisenberg space i.e. periodic minimal surfaces that can be seen as the desingularization of vertical planes intersecting equiangularly. The key point is the construction of a suitable barrier to ensure the convergence of a family of bounded minimal disks. Such a barrier is actually a periodic deformation of a minimal plane with prescribed asymptotic behavior. A consequence of the barrier construction is that the number of disjoint minimal graphs suppoerted on domains is not bounded in Heisenberg space