A semi-analytical, fast and robust method for the propagation of ultra-high-energy cosmic rays (UHECR) under extra-galactic turbulent magnetic fields presented by the authors in previous works is described in detail. Under the assumption of dominant turbulent fields, it is possible to obtain the energy spectrum, as well as the relative contribution from each source, and the angular distribution of arrival directions without the need for 4-D simulations. This method provides an important tool in the quest for the origin of UHECR. A brief overview of the results already obtained with it is presented. The relative contribution to the energy spectrum from sources in different distance shells is elucidated and constraints of a few tens of Mpc, depending on composition, are imposed to the distance to the nearest source. A solid understanding of the arising of the dipole as well as its evolution with energy is built. The importance of a power spectrum analysis in energies detected by experiments is discussed.