Harmonic morphisms are maps $\phi:(M,g)\to (N,h)$ between Riemannian manifolds pulling back harmonic functions on N to harmonic functions on M. They have a non-trivial existence theory and it is therefore interesting to construct manifolds allowing for the existence of solutions to this non-linear problem. Special well-known cases are the holomorphic maps from Kähler manifolds to Riemann surfaces. Investigations have shown that the strong Kähler conditions can be relaxed in interesting cases. In this talk we will construct eight-dimensional Hermitian Lie groups foliated by SU(2)xSU(2)-leaves allowing for the existence of harmonic morphisms.