In The Foundations of Mathematics (p.33), Kenneth Kunen gives the following definitions:
Definition I.8.1 $z$ is a transitive set iff $\forall{y}\in\ z[y\subseteq{z}]$; equivalently, $\forall{xy}[x\in{y}\ \wedge\ y\in{z}\to\ x\in{z}]$.
Definition I.8.2 $z$ is a (von Neumann) ordinal iff $z$ is a transitive set and $z$ is well-ordered by $\in$.
I would like to see an example of a linearly ordered1 (but not well-ordered) transitive set.
(To make this example set most directly comparable to a von Neumman ordinal, I imagine that the order relation should be $\in$.)
1 By linearly ordered I mean that the order relation on the set is irreflexive and satisfies trichotomy (i.e. for all $x, y$ in the relation's field, either $x < y$, $x = y$, or $x > y$).