  We consider a classical interacting dimer model which interpolates between
the square lattice case and the triangular lattice case by tuning a chemical
potential in the diagonal bonds. The interaction energy simply corresponds to
the number of plaquettes with parallel dimers. Using transfer matrix
calculations, we find in the anisotropic triangular case a succession of
different physical phases as the interaction strength is increased: a short
range disordered liquid dimer phase at low interactions, then a critical phase
similar to the one found for the square lattice, and finally a transition to an
ordered columnar phase for large interactions. The existence of the critical
phase is in contrast with the belief that criticality for dimer models is
ascribed to bipartiteness. For the isotropic triangular case, we have
indications that the system undergoes a first order phase transition to an
ordered phase, without appearance of an intermediate critical phase.
