  We present an extensive numerical study of the critical behavior of dimer
models in three dimensions, focusing on the phase transition between Coulomb
and crystalline columnar phases. The case of attractive interactions between
parallel dimers on a plaquette was shown to undergo a continuous phase
transition with critical exponents close to those of the O(N) tricritical
universality class, a situation which is not easily captured by conventional
field theories. That the dimer model is exactly fine-tuned to a highly
symmetric point is a non trivial statement which needs careful numerical
investigation. In this paper, we perform an extensive Monte Carlo study of a
generalized dimer model with plaquette and cubic interactions and determine its
extended phase diagram. We find that when both interactions favor alignment of
the dimers, the phase transition is first order, in almost all cases. On the
opposite, when interactions compete, the transition becomes continuous, with a
critical exponent \eta ~ 0.2. The existence of a tricritical point between the
two regimes is confirmed by simulations on very large size systems and a
flowgram method. In addition, we find a highly-degenerate crystalline phase at
very low temperature in the frustrated regime which is separated from the
columnar phase by a first order transition.
