  We construct non-zero constant mean curvature H surfaces in the product
spaces $\mathbb{S}^2 \times \mathbb{R}$ and $\mathbb{H}^2\times \mathbb{R}$ by
using suitable conjugate Plateau constructions. The resulting surfaces are
complete, have bounded height and are invariant under a discrete group of
horizontal translations. In $\mathbb{S}^2\times\mathbb{R}$ (for any $H > 0$) or
$\mathbb{H}^2\times\mathbb{R}$ (for $H > 1/2$), a 1-parameter family of
unduloid-type surfaces is obtained, some of which are shown to be compact in
$\mathbb{S}^2\times\mathbb{R}$. Finally, in the case of $H = 1/2$ in
$\mathbb{H}^2 \times \mathbb{R}$, the constructed examples have the symmetries
of a tessellation of $\mathbb{H}^2$ by regular polygons.
