  We study the prescribed mean curvature equation for $t$-graphs in a
Riemannian Heisenberg group of arbitrary dimension. We characterize the
existence of classical solutions in a bounded domain without imposing Dirichlet
boundary data, and we provide conditions that guarantee uniqueness. Moreover,
we extend previous results to solve the Dirichlet problem when the mean
curvature is non-constant. Finally, by an approximation technique, we obtain
solutions to the sub-Riemannian prescribed mean curvature equation.
