de Bruijn studies this sum in "On the number of integers $$\le x$$ whose prime factors divide $$n$$", which was published in a 1962 volume of the Illinois J. Math; see
He proves there (see Theorem 1) that $$\sum_{n \le x} \frac{1}{\mathrm{rad}(n)} = \exp((1+o(1)) \sqrt{8\log{x}/\log\log{x}}),$$ as $$x\to\infty$$. Of course, this implies the $$O(x^{\epsilon})$$ bound you were after. However, his proof (which uses a Tauberian theorem of Hardy and Ramanujan) is not as elementary as some others that have been suggested here (but gives a more precise result).