Recently Downarowicz, Huczek, and Zhang proved that every amenable discrete group can be tiled by translates of finitely many approximately invariant finite sets. Using this result I will show that for every free probability-measure-preserving action of an amenable discrete group there are decompositions of the space into finitely many Rokhlin towers with approximately invariant shapes, strengthening a theorem of Ornstein and Weiss. I will then discuss applications to topological dynamics and the study of C*-crossed products.