We investigate the elliptic and the triangular flow of heavy mesons in ultrarelativistic heavy-ion collisions at RHIC and the LHC. The dynamics of heavy quarks is coupled to the locally thermalized and fluid dynamically evolving quark-gluon plasma. The elliptic flow of $D$ mesons and the centrality dependence measured at the LHC is well reproduced for purely collisional and bremsstrahlung interactions. Due to the event-by-event fluctuating initial conditions from the EPOS2 model, the $D$ meson triangular flow is predicted to be nonzero at $\sqrt{s}=200$ GeV and $\sqrt{s}=2.76$ TeV. We study the centrality dependence and quantify the contributions stemming from flow of the light bulk event and the hadronization process. The flow coefficients as response to the initial eccentricities behave differently for heavy mesons than for light hadrons due to their inertia. Higher-order flow coefficients of heavy flavor become important in order to quantify the degree of thermalization.