ANAIS (annual modulation with NaI scintillators) is a dark matter direct detection experiment consisting of 112.5 kg of NaI(Tl) detectors in operation at the Canfranc Underground Laboratory (LSC), in Spain, since August 2017. ANAIS' goal is to confirm or refute in a model independent way the DAMA/LIBRA positive result: an annual modulation in the low-energy detection rate having all the features expected for the signal induced by dark matter particles in a standard galactic halo. This modulation, observed for about 20 years, is in strong tension with the negative results of other very sensitive experiments, but a model-independent comparison is still lacking. By using the same target material, NaI(Tl), such a comparison is more direct and almost independent in dark matter particle and halo models. Here, we present the annual modulation analysis corresponding to three years of ANAIS data (for an effective exposure of 313.95 kg x y), applying a blind procedure, which updates the one developed for the 1.5 years analysis, and later applied to 2 years. The analysis also improves the background modeling in the fitting of the region of interest rates. We obtain for the best fit in the [1-6] keV ([2-6] keV) energy region a modulation amplitude of -0.0034 +/- 0.0042 cpd/kg/keV (0.0003 +/- 0.0037 cpd/kg/keV), supporting the absence of modulation in our data, and incompatible with the DAMA/LIBRA result at 3.3 (2.6) sigma, for a sensitivity of 2.5 (2.7) sigma. Moreover, we include two complementary analyses: a phase-free annual modulation search and the exploration of the possible presence of a periodic signal at other frequencies. Finally, we carry out several consistency checks of our result, and we update the ANAIS-112 projected sensitivity for the scheduled 5 years of operation.