A search is performed for a heavy particle decaying into different-flavor, dilepton pairs (eμ, eτ or μτ), using 36.1 fb−1 of proton-proton collision data at s=13 TeV collected in 2015–2016 by the ATLAS detector at the Large Hadron Collider. No excesses over the Standard Model predictions are observed. Bayesian lower limits at the 95% credibility level are placed on the mass of a Z′ boson, the mass of a supersymmetric τ-sneutrino, and on the threshold mass for quantum black-hole production. For the Z′ and sneutrino models, upper cross-section limits are converted to upper limits on couplings, which are compared with similar limits from low-energy experiments and which are more stringent for the eτ and μτ modes.