On the existence of smooth varieties of any dimension carrying no Ulrich bundles for some very ample line bundle

Using some recent examples of Anghel in dimension $2$, we prove that for any $n\ge 3$ there exist many smooth $n$-dimensional varieties $X$ with a very ample line bundle $H$, such that there is no Ulrich bundle for $(X,H)$. Moreover, we find many new examples in dimension $2$.