In 2-dimensional topology, quotients of polygons by edge-pairings provide a rich source of examples of closed, connected, orientable surfaces. In fact, they provide all such examples. The 3-dimensional analogue of an edge-pairing of a polygon is a face-pairing of a faceted 3-ball. Unfortunately, quotients of faceted 3-balls by face-pairings rarely provide us with examples of 3-manifolds due to singularities that arise at the vertices. However, any face-pairing of a faceted 3-ball may be slighted modified so that its quotient is a genuine manifold, i.e. free of singularities. The modified face-pairing is called a twisted face-pairing. It is natural to ask which closed, connected, orientable 3-manifolds may be obtained as quotients of twisted face-pairings. In this paper, we focus on a special class of face-pairings called cycle-free twisted face-pairings and give description of their quotient spaces in terms of integer weighted graphs. We use this description to prove that most spherical 3-manifolds can be obtained as quotients of cycle-free twisted face-pairings, but the Poincaré homology 3-sphere cannot.