The quantitative extraction of quark-gluon plasma (QGP) properties from heavy-ion data, such as its specific shear viscosity η /s, typically requires comparison to viscous hydrodynamic or "hybrid" hydrodynamics+transport simulations. In either case, one has to convert the fluid to hadrons, yet without additional theory input the conversion is ambiguous for dissipative fluids. Here, shear viscous phase-space corrections calculated using linearized transport theory are applied in Cooper-Frye

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... eout to quantify the effects on anisotropic flow coefficients v_n(p_T) at both RHIC and LHC energies. Expanding upon our previous flow harmonics studies [1,2], we calculate pion and proton v_2(p_T), v_4(p_T), and v_6(p_T). Unlike in Ref. [1], we incorporate a hadron gas that is chemically frozen below a temperature of 175 MeV, and use hypersurfaces from realistic viscous hydrodynamic simulations. With additive quark model cross sections and relative phase-space corrections with p^3/2 momentum dependence, rather than the quadratic Grad form, we find at moderately high transverse momentum noticeably higher v_4(p_T) and v_6(p_T) for protons than for pions. In addition, the value of η /s deduced from elliptic flow data differs by nearly 50% from the value extracted using the naive "democratic Grad" form of freeze-out distributions. To facilitate the use of the self-consistent viscous corrections calculated here in hydrodynamic and hybrid calculations, we also present convenient parameterizations of the corrections for the various hadron species (cf. Table I).