  We investigate the threshold $p_{\vec H}=p_{\vec H}(n)$ for the Ramsey-type
property $G(n,p)\to \vec H$, where $G(n,p)$ is the binomial random graph and
$G\to\vec H$ indicates that every orientation of the graph $G$ contains the
oriented graph $\vec H$ as a subdigraph. Similarly to the classical Ramsey
setting, the upper bound $p_{\vec H}\leq Cn^{-1/m_2(\vec H)}$ is known to hold
for some constant $C=C(\vec H)$, where $m_2(\vec H)$ denotes the maximum
$2$-density of the underlying graph $H$ of $\vec H$. While this upper bound is
indeed the threshold for some $\vec H$, this is not always the case. We obtain
examples arising from rooted products of orientations of sparse graphs (such as
forests, cycles and, more generally, subcubic $\{K_3,K_{3,3}\}$-free graphs)
and arbitrarily rooted transitive triangles.
