Measurements of the anisotropic flow coefficients v(2){Psi(2)}, v(3){Psi(3)}, v(4){Psi(4)}, and v(4){Psi(2)} for identified particles (pi(+/-), K-+/-, and p + (p) over bar) at midrapidity, obtained relative to the event planes Psi(m) at forward rapidities in Au + Au collisions at root s(NN) = 200 GeV, are presented as a function of collision centrality and particle transverse momenta p(T). The v(n) coefficients show characteristic patterns consistent with hydrodynamical expansion of the matter produced in the collisions. For each harmonic n, a modified valence quark-number N-q scaling [plotting v(n){Psi(m)}/(N-q)(n/2) versus transverse kinetic energies (KET)/N-q] is observed to yield a single curve for all the measured particle species for a broad range of KET. A simultaneous blast-wave model fit to the observed v(n){Psi(m)}(p(T)) coefficients and published particle spectra identifies radial flow anisotropies rho(n){Psi(m)} and spatial eccentricities s(n){Psi(m)} at freeze-out. These are generally smaller than the initial-state participant-plane geometric eccentricities epsilon(n){Psi(PP)(m)} as also observed in the final eccentricity from quantum interferometry measurements with respect to the event plane.