Even though this question is old, I'd like to give what I regard as a very beautiful solution. It is different from the others in that the circles used are not round (but they are unlinked). First observe that the circles $x^2 + y^2 = r^2$, $z = c$, for $r geq 1$ and $c$ any real number, decompose all of $Bbb{R}^3$ except an open cylinder into circles. At first glance, this seems to have accomplished nothing, since the open cylinder is homeomorphic to $Bbb{R}^3$, so we have reduced the original problem to an equivalent problem. However, look at the left-hand figure of the included image, which shows an open cylinder embedded as a U shape, with the ends going to infinity in the same direction. Since this is just a deformation of the original embedding, we can decompose the complement into circles. To handle the interior, embed an open cylinder into it, as shown in the right-hand figure. We can decompose the complement of the smaller U-shaped cylinder into circles. We continue in this way, making sure that the embedded cylinders go off to infinity, so that every point of $Bbb{R}^3$ is included at some finite stage.
It seems like we have never really solved the problem, but instead have just pushed it away so much that it vanishes into thin air!
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