Let \(U\subset \mathbb{R}^{2m}\) be an open set and consider on \(U\) a non- Kähler, Hermitian structure \(J\). The authors study the existence of harmonic maps \(\phi: (U, J)\to \mathbb{C}\) by extending the results from the second author [Int. J. Math. 3, No. 3, 415-439 (1992; Zbl 0763.53051)] to higher dimensions. They find many interesting locally and globally defined harmonic morphisms from open sets of \(\mathbb{R}^{2m}\), which are holomorphic with respect to non-Kähler structures. There are important differences between the case \(m= 2\) and the higher-dimensional cases.