The leaves of the foliation are regular. So, the space of leaves $X$ induces a submersion $\pi: R^3\rightarrow X$ with leaves of $R.$ Using the standard homotopy sequence, we see that $X$ is simply-connected. Furthermore, $X$ is path-connected since any two leaves can be connected by a path of leaves in the foliation.
We can actually define an embedding $\phi: X\rightarrow R^3$ by letting $\phi(l)$ be the point on the leaf $l\in X$ that is closest to the origin in $R^3.$ This gives us two things: a vector space structure on each leaf by using $\phi(l)$ as the origin and an orientation of $X$ (since only orientable simply-connected surfaces are embeddable in $R^3$). We use the orientation of $X$ to give an orientation on each leaf.
Thus, we have a 1-dimensional orientable vector bundle over a simply-connected space $X$, which means that the vector bundle is trivial. Hence, all smooth foliations with leaves of lines on $R^3$ are diffeomorphic.
I rather wanted to say all continuous foliations of such sort are homeomorphic, but I'm nervous about the embedding of $X.$