  We use the motivic obstruction to stable rationality introduced by Shinder
and the first-named author to establish several new classes of stably
irrational hypersurfaces and complete intersections. In particular, we show
that very general quartic fivefolds and complete intersections of a quadric and
a cubic in $\mathbb P^6$ are stably irrational. An important new ingredient is
the use of tropical degeneration techniques.
