since Henry started the shameless self-promotion, let me also do so... Given any group $\pi$ one can study the corresponding Alexander polynomial $\Delta_\pi$ which lies in the group ring of $H:=H_1(\pi;\Bbb{Z})/\mbox{torsion}$. If $\pi$ is the fundamental group of a closed 3-manifold, then the Alexander polynomial $\Delta_{\pi}$ is symmetric and the one-variable specializations have even degree. (see F, Kim, Kitayama: Poincaré duality and degrees of twisted Alexander polynomials)

The symmetry holds also if $\pi$ is a 3-dimensional Poincare duality group, but I am not sure whether the degree condition holds in that case.

The advantage is that this condition can be checked easily, and by checking it for finite index subgroups one gets even more necessary conditions. I would guess that in practice this is a very effective way for weeding out non 3-manifold groups.

At least it allowed me to make the right bet on Ryan's example...