  We prove that for forms of U(3) which are compact at infinity and split at
places dividing a prime p, in generic situations the Serre weights of a mod p
modular Galois representation which is irreducible when restricted to each
decomposition group above p are exactly those previously predicted by the third
author. We do this by combining explicit computations in p-adic Hodge theory,
based on a formalism of strongly divisible modules and Breuil modules with
descent data which we develop in the paper, with a technique that we call
"weight cycling".
