  We construct new complex-valued harmonic morphisms from Euclidean spaces from
functions which are holomorphic with respect to Hermitian structures. In
particular, we give the first global examples of complex-valued harmonic
morphisms from ${\bf R}^n$ for each $n>4$ which do not arise from a K\"ahler
structure; it is known that such examples do not exist for $n \leq 4$.
