  A cyclic ordering of the points in a Mendelsohn triple system of order $v$
(or MTS$(v)$) is called a sequencing. A sequencing $D$ is $\ell$-good if there
does not exist a triple $(x,y,z)$ in the MTS$(v)$ such that (1) the three
points $x,y,$ and $z$ occur (cyclically) in that order in $D$; and (2)
$\{x,y,z\}$ is a subset of $\ell$ cyclically consecutive points of $D$. In this
paper, we prove some upper bounds on $\ell$ for MTS$(v)$ having $\ell$-good
sequencings and we prove that any MTS$(v)$ with $v \geq 7$ has a $3$-good
sequencing. We also determine the optimal sequencings of every MTS$(v)$ with $v
\leq 10$.
