  We show that Veronese varieties of dimension $n \ge 4$ do not carry any
Ulrich bundles of rank $r \le 3$. In order to prove this, we prove that a
Veronese embedding of a complete intersection of dimension $m \ge 4$, which if
$m=4$ is either $\mathbb P^4$ or has degree $d \ge 2$ and is very general and
not of type $(2), (2,2)$, does not carry any Ulrich bundles of rank $r \le 3$.
