  The incommensurate magnetic response observed in normal-state cuprate perovskites is interpreted based on the projection operator formalism and the t-J model of Cu-O planes. In agreement with experiment the calculated dispersion of maxima in the susceptibility has the shape of two parabolas with upward and downward branches which converge at the antiferromagnetic wave vector. The maxima are located at the momenta $({1/2},{1/2}\pm\delta)$, $({1/2}\pm\delta,{1/2})$ and at $({1/2}\pm\delta,{1/2}\pm\delta)$, $({1/2}\pm\delta,{1/2}\mp\delta)$ in the lower and upper parabolas, respectively. The upper parabola reflects the dispersion of magnetic excitations of the localized Cu spins, while the lower parabola arises due to a dip in the spin-excitation damping at the antiferromagnetic wave vector. For moderate doping this dip stems from the weakness of the interaction between the spin excitations and holes near the hot spots. The frequency dependence of the susceptibility is shown to depend strongly on the hole bandwidth and damping and varies from the shape observed in YBa$_2$Cu$_3$O$_{7-y}$ to that inherent in La$_{2-x}$Sr$_x$CuO$_4$. 
Incommensurate spin dynamics in underdoped cuprate perovskites