  We show how the techniques of Voevodsky's proof of the Milnor conjecture and
the Voevodsky- Rost proof of its generalization the Bloch-Kato conjecture can
be used to study counterexamples to the classical L\"uroth problem. By
generalizing a method due to Peyre, we produce for any prime number l and any
integer n >= 2, a rationally connected, non-rational variety for which
non-rationality is detected by a non-trivial degree n unramified \'etale
cohomology class with l-torsion coefficients. When l = 2, the varieties that
are constructed are furthermore unirational and non-rationality cannot be
detected by a torsion unramified \'etale cohomology class of lower degree.
