In this article1, the authors prove that not only can you partition $R^3$ into congruent circles, but you can do so into unlinked congruent circles. They also prove a variety of other similar results: $R^3$ can be partitioned into isometric copies of any family of continuum many real analytic curves. And they consider the question in higher dimensions, and also the role of AC in the proofs: for example, in $R^3$ no AC is needed for circles, if different sizes are allowed.
1M. Jonsson and J. Wästlund: Partition of $R^3$ into curves, Mathematica Scandinavica 83 (1998) 192-204; JSTOR, author's website
