A dynamical 2-complex is a 2-complex equipped with a set of combinatorial properties which allow one to define non-singular semi-flows on the complex. After giving a combinatorial characterization of the dynamical 2-complexes which define hyperbolic attractors when embedded in compact 3-manifolds, we give an effective criterion for the existence of cross sections to the semi-flows on these 2-complexes. In the embedded case, this gives an effective criterion of existence of cross sections to the associated hyperbolic attractors. We present a similar criterion for boundary-tangent flows on compact 3-manifolds which are constructed by means of our dynamical 2-complexes.