Using low-energy projection of the one-band t-t′-t′′ Hubbard model we derive an effective spin Hamiltonian and its spin-wave expansion to order 1/S. We fit the spin-wave dispersion of several parent compounds to the high-temperature superconducting cuprates La2CuO4, Sr2CuO2Cl2, and Bi2Sr2YCu2O8. Our accurate quantitative determination of the one-band Hubbard model parameters allows prediction and comparison to experimental results. Among those we discuss the two-magnon Raman peak line shape, the K-edge resonant inelastic x-ray scattering 500-meV peak, and the high-energy kink in the angle-resolved photoemission spectroscopy quasiparticle dispersion, also known as the waterfall feature.