A search for the decay of neutral, weakly interacting, long-lived particles using data collected by the ATLAS detector at the LHC is presented. This analysis uses the full data set recorded in 2012: 20.3 fb-1 of proton-proton collision data at s=8 TeV. The search employs techniques for reconstructing decay vertices of long-lived particles decaying to jets in the inner tracking detector and muon spectrometer. Signal events require at least two reconstructed vertices. No significant excess of events over the expected background is found, and limits as a function of proper lifetime are reported for the decay of the Higgs boson and other scalar bosons to long-lived particles and for Hidden Valley Z′ and Stealth SUSY benchmark models. The first search results for displaced decays in Z′ and Stealth SUSY models are presented. The upper bounds of the excluded proper lifetimes are the most stringent to date. © 2015 CERN, for the ATLAS Collaboration. Published by the American Physical Society under the terms of the «http://creativecommons.org/licenses/by/3.0/» Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.