As pointed out in a comment by Richard Kent to Allen Knutson's answer, the problem is a bit more subtle than it may appear. In order to prove that the homeomorphism problem for compact 4-manifolds, say in the topological category, is recursively unsolvable, it is not enough to know that (1) every finitely presented group can be realized as the fundamental group of some compact 4-manifold, and (2) the isomorphism problem for finitely presented groups is recursively unsolvable.
Instead, what you do is give a construction which to any finite presentation $< S | P >$ of a group associates a 4-manifold $M(S,P)$ in such a way that $\pi_1(M(S,P))$ is isomorphic to the group defined by the presentation $< S | P >$, and moreover such two manifolds are homeomorphic if and only if they have isomorphic fundamental groups.
Then you have constructed a class of 4-manifolds for which the homeomorphism problem is equivalent to the isomorphism problem for finitely presented groups, and therefore unsolvable.
About "geometrization for manifolds of dimension 4 or higher", well as far as I know there is no theorem which says it is impossible. It depends on what you mean by `geometric structure', and what you want those structures to do for you.