  We prove that there exists a universal constant $r_3$ such that if $X$ is a
smooth projective threefold over $\mathbb{C}$ with non-negative Kodaira
dimension, then the linear system $|r K_X|$ admits a fibration that is
birational to the Iitaka fibration as soon as $r \geq r_3$ and sufficiently
divisible. This gives an affirmative answer to a conjecture of Hacon and
McKernan in the case of threefolds. Viehweg and Zhang have posted a stronger
result along these lines using different methods.
