There is an analogy to surfaces in a sense. For 3-manifolds that fibre over surfaces there is a complete answer. For a variety of Seifert-fibred manifolds there are complete answers -- but not all. For example, Seifert-fibred homology spheres are still problematic. The preprint that Ian linked to in his comments has much more results of this kind in it.
At present, in summary:
1) We likely do not have a complete set of invariants that obstruct embedding into $\mathbb R^4$.
2) We appear to be far from knowing all the "natural" constructions of embeddings of 3-manifolds into $\mathbb R^4$ for the manifolds that are known to embed.
It is quite possible there are elements of formal logic obstructing both 1 and 2.
For example, if a compact boundaryless connected 3-manifold embeds in $S^4$ it separates it into two components. It is possible that one or even both of these components has a fundamental group with an unsolvable word problem. This would restrict the kinds of techniques one could use for creating obstructions in (1).
edit: I see Agol and Freedman's paper on this topic as connected to this last concern. 2-manifolds in $S^3$ have the Fox re-embedding theorem. So you could hope for some nice re-embedding theorems for $3$-manifolds in $S^4$. You shouldn't expect too nice a re-embedding theorem in $S^4$, since the tool that makes Fox's theorem work is Dehn's lemma, and the analogies to Dehn's lemma in 4-manifold theory are generally not true.