The production of the prompt charmed mesons D0, D+, and D*+ relative to the reaction plane was measured in Pb-Pb collisions at a center-of-mass energy per nucleon-nucleon collision of √sNN=2.76TeV with the ALICE detector at the CERN Large Hadron Collider. D mesons were reconstructed via their hadronic decays at central rapidity in the transverse-momentum (p_T) interval 2–16 GeV/c. The azimuthal anisotropy is quantified in terms of the second coefficient v2 in a Fourier expansion of the D-meson azimuthal distribution and in terms of the nuclear modification factor R_AA, measured in the direction of the reaction plane and orthogonal to it. The v2 coefficient was measured with three different methods and in three centrality classes in the interval 0%–50%. A positive v2 is observed in midcentral collisions (30%–50% centrality class), with a mean value of 0.204+0.099−0.036 (tot. unc.) in the interval 2<pT<6GeV/c, which decreases towards more central collisions (10%–30% and 0%–10% classes). The positive v2 is also reflected in the nuclear modification factor, which shows a stronger suppression in the direction orthogonal to the reaction plane for midcentral collisions. The measurements are compared to theoretical calculations of charm-quark transport and energy loss in high-density strongly interacting matter at high temperature. The models that include substantial elastic interactions with an expanding medium provide a good description of the observed anisotropy. However, they are challenged to simultaneously describe the strong suppression of high-pT yield of D mesons in central collisions and their azimuthal anisotropy in noncentral collisions.