propagation of classical fields

it is a well-known fact that general relativity requires at least 4 dimensions to work, otherwise the Weyl-tensor would be zero, or the field would not have dynamical degrees of freedom: what is left from this relation in classical gravity with \(\Delta\Phi=4\pi G\rho\)? would it work in any number of dimensions, and why does it only start from \(n=3\) on to be sensible?