A search for heavy right-handed Majorana or Dirac neutrinos N_{R} and heavy right-handed gauge bosons W_{R} is performed in events with a pair of energetic electrons or muons, with the same or opposite electric charge, and two energetic jets. The events are selected from pp collision data with an integrated luminosity of 36.1 fb⁻¹ collected by the ATLAS detector at TeV. No significant deviations from the Standard Model are observed. The results are interpreted within the theoretical framework of a left-right symmetric model and lower limits are set on masses in the heavy right-handed W boson and neutrino mass plane. The excluded region extends to TeV for both Majorana and Dirac N_{R} neutrinos.