Homology sphere with $\mathbb{R}^3$ as the universal cover

In a sense, most $3$-manifolds have universal cover $R^3$. In particular, this is the case for hyperbolic $3$-manifolds. And there do exist integer homology spheres which are hyperbolic. Two explicit examples I found by googling: Auckly: Surgery numbers of 3-manifolds: a hyperbolic example and Hom, Lidman: Surgery obstructions and hyperbolic integer homology spheres. It should be possible to get more examples from SnapPy.

In fact, this is true for any closed connected 3-manifold with aspherical fundamental group. More generally, the universal cover of a closed 3-manifold is $S^3$ punctured at $0, 1, 2$ points or a tame Cantor set.

Aspherical homology spheres abound, for example $1/n$ surgery on a knot for $n$ sufficiently large. Other examples are Brieskorn spheres $\Sigma(p,q,r)$ with $p,q,r$ pairwise relatively prime and $1/p+1/q+1/r \leq 1$.

There are several good answers but I thought I'd chuck in one more. Prior to all of the important results on geometrization mentioned above, Waldhausen had shown that any Haken manifold (any 2-sphere bounds a ball and the manifold contains an incompressible surface) has universal cover $\mathbb{R}^3$. Simple examples of Haken homology spheres are obtained by gluing two non-trivial knot complements in such a way that meridian and longitude are interchanged.

Piotr also asked about higher dimensional examples. These are harder to come by. In dimension $4$ there is "Some examples of aspherical 4-manifolds that are homology 4-spheres" by Ratcliffe and Tschantz, Topology Volume 44, Issue 2, March 2005, Pages 341-350. I don't know about dimensions higher than that.