Update:
My memory was quite blurry about this when I originally answered.
See Gonzáles-Acuña, Gordon, Simon, ``Unsolvable problems about higher-dimensional knots and related groups,'' L’Enseignement Mathématique (2) 56 (2010), 143-171.
They prove that any finitely presented group is a subgroup of the fundamental group of the complement of a closed orientable surface in the $4$-sphere, which is much better than I reported.
Original answer:
You most likely would like a finitely presented group, but this might be of interest anyway:
Let $S$ be a recursively enumerable non-recursive subset of the natural numbers and consider the group
$\langle \ a,b,c,d \ | \ a^iba^{-i} = c^idc^{-i} \ \mathrm{for}\ i \in S \rangle$
This has unsolvable word problem. See page 110 of Chiswell's book "A course in formal languages, automata and groups" available on google books (I think it's also in Baumslag's "Topics in Combinatorial Group Theory" but all my books are in boxes at the moment.)
This should be the fundamental group of the complement of a noncompact surface in $\mathbb{R}^4$. You do this in the usual way by beginning with the trivial link on four components and then drawing the movie of the surface in $\mathbb{R}^4$, band summing at each stage to make the conjugates of $b$ and $d$ equal.
I think you end up with a knotted disjoint union of planes. I remember doing this at some point in graduate school when C. Gordon asked me if there were any compact surfaces in the $4$-sphere whose complements have groups with unsolvable word problem.
Best Answer
See edit at bottom for further information answering the question in all dimensions.
In all odd dimensions $2k -1 > 3$, there are non-homeomorphic homology spheres with fundamental group G = the binary icosahedral group (fundamental group of the Poincaré homology sphere). This follows from basic surgery theory; the Wall group $L_{2k}(G)$ contains a subgroup of the form $\mathbb{Z}^n$ for some $n$ related to the number of irreducible complex representations of $G$. This subgroup is detected by the so-called multisignature, as described in Wall's book on surgery theory.
Choose one homology sphere $M^{2k-1}$ with $\pi_1(M) = G$. For instance, you can repeatedly spin the Poincaré sphere, where spinning $P$ means doing surgery on the obvious $S^1$ in $S^1\times P$. For any $M$ with $\pi_1(M) = G$, there's an invariant originally described by Atiyah and Singer. A priori, it's a smooth invariant, but is known to be a homeomorphism invariant. Roughly speaking, one knows that for some $d \in \mathbb{N}$, the manifold $d\cdot M$ is the boundary of a $2k$-manifold $X$ with $\pi_1(X) = G$. Then $X$ has a collection of equivariant signatures (associated to the action of $G$ on the universal cover of $X$), known collectively as the multisignature. The multisignature (divided by $d$) is a topological invariant. (Technically, this is only well-defined up to a choice of isomorphism $\pi_1(X) \to G$ but this is readily dealt with.)
Now, the group $L_{2k}(G)$ acts on the structure set of $M$, as described in Wall's book. The effect of the action is to change the multisignature, and hence it changes the homeomorphism type of $M$. In this construction, you not only preserve the fundamental group, but also the (simple) homotopy type of $M$.
It's possible that acting on an even-dimensional homology sphere $M^{2k}$ with fundamental group $G$ by elements of $L_{2k+1}(G)$ could change the homeomorphism type. But odd dimensional $L$-groups are much harder, and you'd need some serious expertise to see what the effect should be. I rather suspect that you could do something simpler, and change the homology with local coefficients or something like that.
Addendum: There are two papers of Alex Suciu that answer this question in dimensions at least 4. In "Homology 4-spheres with distinct k-invariants," Topology and its Applications 25 (1987) 103-110, he gives examples of homology 4-spheres with the same $\pi_1$ and $\pi_2$ that are not homotopy equivalent. In "Iterated spinning and homology spheres", Trans AMS 321 (1990) he constructs homology n-spheres in dimensions $n \geq 5$ with the same property. Combined with the remarks above about dimension 3, this answers your question.