  We consider translating solitons to flows by positive powers $\alpha$ of the
Gaussian curvature -- called $K^\alpha$-flows -- in Riemannian products
$M\times\mathbb R.$ We prove that, when $M$ is the Euclidean space $\mathbb
R^n,$ the sphere $\mathbb S^n,$ or one of the hyperbolic spaces
$\mathbb{H}_{\mathbb F}^m,$ there exist complete rotational translating
solitons to $K^\alpha$-flow in $M\times\mathbb R$ for certain values of
$\alpha.$
