For d=3 the homotopy groups can be pretty elaborate. Consider the connect-sum of some lens spaces. The universal cover embeds in S^3 as the complement of a cantor set (except for a few degenerate cases where you have RP^3 summands). So the homotopy-groups are pretty complicated (\pi_2 is finitely generated over \pi_1). You could probably make an argument that this is about the worst thing that can happen for the homotopy-groups of 3-manifolds.
You might want to phrase your question as a question about the Postnikov towers of manifolds. Eilenberg-Maclane spaces are rarely compact boundaryless manifolds.