A search is performed for a heavy particle decaying into different-flavor, dilepton pairs ($e\mu$, $e\tau$ or $\mu \tau$), using 36.1 fb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$ 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 $Z\prime$ boson, the mass of a supersymmetric $\tau$-sneutrino, and on the threshold mass for quantum black-hole production. For the $Z\prime$ 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 $e\tau$ and $\mu \tau$ modes