The Radiroot package for GAP by Andreas Distler solves by radicals polynomials with solvable Galois group. The package uses the algorithm described in Distler's paper "Ein Algorithmus zum Lösen einer Polynomgleichung durch Radikal". As you would guess from the title, the paper is in German, which I cannot read. Is there an English (or Russian) translation of this paper?
I've checked around, but can't seem to find a translated version of the paper, be it English or Russian. I did Google it. As a last resort, you can try Google's "translate" option: it rendered a decent translation, though formatting suffers.
Perhaps your best bet would be to contact Andrew Distler directly?