In the present paper we introduce and study a class of three-dimensional manifolds endowed with the action of the group $\mathbb Z_2^3$ whose orbit space is a simple convex polytope. These manifolds originate from three-dimensional polytopes whose faces allow a coloring into three colors with the help of the construction used for studying quasitoric manifolds. For such manifolds we prove the existence of an equivariant embedding into Euclidean space $\mathbb R^4$. We also describe the action on the set of operations of the equivariant connected sum and the equivariant Dehn surgery. We prove that any such manifold can be obtained from a finitely many three-dimensional tori with the canonical action of the group $\mathbb Z_2^3$ by using these operations.