Let $Q$ be an euclidean quiver. Using friezes in the sense of
Assem-Reutenauer-Smith, we provide an algorithm for computing the (canonical)
cluster character associated to any object in the cluster category of $Q$. In
particular, this algorithm allows to compute all the cluster variables in the
cluster algebra associated to $Q$. It also allows to compute the sum of the
Euler characteristics of the quiver grassmannians of any module $M$ over the
path algebra of $Q$.