  The Tur\'an density of an $r$-uniform hypergraph $\mathcal{H}$, denoted
$\pi(\mathcal{H})$, is the limit of the maximum density of an $n$-vertex
$r$-uniform hypergraph not containing a copy of $\mathcal{H}$, as $n \to
\infty$.
  Denote by $\mathcal{C}_{\ell}$ the $3$-uniform tight cycle on $\ell$
vertices. Mubayi and R\"odl gave an ``iterated blow-up'' construction showing
that the Tur\'an density of $\mathcal{C}_5$ is at least $2\sqrt{3} - 3 \approx
0.464$, and this bound is conjectured to be tight. Their construction also does
not contain $\mathcal{C}_{\ell}$ for larger $\ell$ not divisible by $3$, which
suggests that it might be the extremal construction for these hypergraphs as
well. Here, we determine the Tur\'an density of $\mathcal{C}_{\ell}$ for all
large $\ell$ not divisible by $3$, showing that indeed $\pi(\mathcal{C}_{\ell})
= 2\sqrt{3} - 3$. To our knowledge, this is the first example of a Tur\'an
density being determined where the extremal construction is an iterated blow-up
construction.
  A key component in our proof, which may be of independent interest, is a
$3$-uniform analogue of the statement ``a graph is bipartite if and only if it
does not contain an odd cycle''.
