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

https://projecteuclid-org.proxy-remote.galib.uga.edu/euclid.ijm/1255631814

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).