The Green function of linear time-domain hydrodynamics has been recently proven to satisfy a simple fourth order ordinary differential equation (ODE). As a first application of this remarkable property, the present paper shows how it can be used to evaluate the kernel of the convolution integrals in the BEM solution of seakeeping problems, in the linear time-domain approach. The Green function is obtained by integrating these ODE from one time-step to another instead of being computed by the classical routines. Several Runge-Kutta algorithms were tested and compared; it is shown how an optimal time step can be derived from stability analysis. The CPU-time savings brought by this new method are then evaluated. Numerical results are given for three test cases: the heaving hemisphere, a standard series-60 ship hull, and a TLP offshore platform. Impulse response functions are calculated and Fourier transformed to recover the classical frequency domain hydrodynamic coefficients as a check for validity and accuracy.