  Let F be a nonarchimedean local field of odd residual characteristic p. We
classify finite-dimensional simple right modules for the pro-p-Iwahori-Hecke
algebra $\mathcal{H}_C(G,I(1))$, where G is the unramified unitary group
U(2,1)(E/F) in three variables. Using this description when C is the algebraic
closure of $\mathbb{F}_p$, we define supersingular Hecke modules and show that
the functor of I(1)-invariants induces a bijection between irreducible
nonsupersingular mod-p representations of G and nonsupersingular simple right
$\mathcal{H}_C(G,I(1))$-modules. We then use an argument of Paskunas to
construct supersingular representations of G.
