The elliptic, triangular, quadrangular and pentagonal anisotropic flow coefficients for π±, K± and p + p ¯ in Pb-Pb collisions at sNN=2.76 TeV were measured with the ALICE detector at the Large Hadron Collider. The results were obtained with the Scalar Product method, correlating the identified hadrons with reference particles from a different pseudorapidity region. Effects not related to the common event symmetry planes (non-flow) were estimated using correlations in pp collisions and were subtracted from the measurement. The obtained flow coefficients exhibit a clear mass ordering for transverse momentum (pT) values below ≈ 3 GeV/c. In the intermediate pT region (3 < pT< 6 GeV/c), particles group at an approximate level according to the number of constituent quarks, suggesting that coalescence might be the relevant particle production mechanism in this region. The results for pT< 3 GeV/c are described fairly well by a hydrodynamical model (iEBE-VISHNU) that uses initial conditions generated by A Multi-Phase Transport model (AMPT) and describes the expansion of the fireball using a value of 0.08 for the ratio of shear viscosity to entropy density (η/s), coupled to a hadronic cascade model (UrQMD). Finally, expectations from AMPT alone fail to quantitatively describe the measurements for all harmonics throughout the measured transverse momentum region. However, the comparison to the AMPT model highlights the importance of the late hadronic rescattering stage to the development of the observed mass ordering at low values of pT and of coalescence as a particle production mechanism for the particle type grouping at intermediate values of pT for all harmonics.[Figure not available: see fulltext.]