Sakurai says that the propagator is simply the Green's function for the time-dependent wave equation satisfying
$$\left [ -\frac{\hbar^2}{2m} \triangledown ''^2+V(\mathbf{x''})-ih\frac{\partial }{\partial t}\right ]K(\mathbf{x''},t;\mathbf{x'},t_0)=-i\hbar\delta ^3(\mathbf{x''}-\mathbf{x'})\delta (t-t_0)$$
with the boundary condition
$$K(\mathbf{x''},t;\mathbf{x'},t_0)=0$$
for $t<t_0$
I don't have any idea about where the $-i\hbar\delta ^3(\mathbf{x''}-\mathbf{x'})\delta (t-t_0)$ term comes from, and the propagator must be equal to zero when $t<t_o$.