Let $X$ be a variety defined over an algebraically closed field of characteristic 0 and let $\phi\colon X\to X$ be a birational automorphism. The Medvedev-Scanlon conjecture predicts when there is a rational point of X with dense orbit under $\phi$. We prove their conjecture in positive Kodaira dimension and then, contingent on conjectures in the Minimal Model Program, prove the conjecture for certain minimal threefolds of Kodaira dimension 0. This is joint work with Jason Bell, Dragos Ghioca, and Zinovy Reichstein.