  We prove that any $3$-uniform hypergraph whose minimum vertex degree is at
least $\left(\frac{5}{9} + o(1) \right)\binom{n}{2}$ admits an almost-spanning
tight cycle, that is, a tight cycle leaving $o(n)$ vertices uncovered. The
bound on the vertex degree is asymptotically best possible. Our proof uses the
hypergraph regularity method, and in particular a recent version of the
hypergraph regularity lemma proved by Allen, B\"ottcher, Cooley and Mycroft.
