Hydrogen Balmer lines are commonly used as spectroscopic effective temperature diagnostics of late-type stars. However, reliable inferences require accurate model spectra, and the absolute accuracy of classical methods that are based on one-dimensional (1D) hydrostatic model atmospheres and local thermodynamic equilibrium (LTE) is still unclear. To investigate this, we carry out 3D non-LTE calculations for the Balmer lines, performed, for the first time, over an extensive grid of 3D hydrodynamic STAGGER model atmospheres. For H{alpha}, H{beta}, and H{gamma} we find significant 1D non-LTE versus 3D non-LTE differences (3D effects): the outer wings tend to be stronger in 3D models, particularly for H{gamma}, while the inner wings can be weaker in 3D models, particularly for H{alpha}. For H{alpha}, we also find significant 3D LTE versus 3D non-LTE differences (non-LTE effects): in warmer stars (Teff~~6500K) the inner wings tend to be weaker in non-LTE models, while at lower effective temperatures (Teff~~4500K) the inner wings can be stronger in non-LTE models; the non-LTE effects are more severe at lower metallicities. We test our 3D non-LTE models against observations of well-studied benchmark stars. For the Sun, we infer concordant effective temperatures from H{alpha}, H{beta}, and H{gamma}; however the value is too low by around 50K which could signal residual modelling shortcomings. For other benchmark stars, our 3D non-LTE models generally reproduce the effective temperatures to within 1{sigma} uncertainties. For H{alpha}, the absolute 3D effects and non-LTE effects can separately reach around 100K, in terms of inferred effective temperatures. For metal-poor turn-off stars, 1D LTE models of H{alpha} can underestimate effective temperatures by around 150K. Our 3D non-LTE model spectra are publicly available, and can be used for more reliable spectroscopic effective temperature determinations.