Abstract <p>Masses of the ground and orbitally excited states of triply heavy tetraquarks with one open heavy flavor are calculated within the relativistic quark model, based on the quasipotential approach and QCD. Tetraquark is treated as a bound state of the nonobservable nonpointlike diquark and antidiquark. The relativistic quasipotential equation takes into account all relativistic corrections (both spin-dependent and spin-independent) and the internal structure of the diquarks via the insertion of calculated diquark–gluon form factors. The obtained masses of tetraquarks are compared with the strong fall-apart decay thresholds into a pair of heavy and heavy-light mesons. The states that lie slightly above or under such thresholds and thus could be observed as narrow resonances in other decay modes are identified.</p>