The proposed methodology is a time-domain panel method where the transient Green functions (GFs) used for the estimation and implementation of the free-surface effects on the vessel's motions are estimated assuming constant low lateral speed, instead of the common practice zero-speed influence functions. The main difference between the proposed method and standard practice lies in the use of the proposed scheme on a typical zero-speed problem, as the solution scheme is conditioned to the mean drift velocity instead of zero-speed. Furthermore, in this way, a significant improvement in accuracy accompanied by a large reduction in computational times is also achieved. The low lateral-speed GFs are computed for a speed similar to the one that the vessel is expected to drift. Estimation of this speed can be initially based on model tests or by means of an iterative process where we start the computations with the zero-speed GFs and adjust the latter's input speed according to the calculated drift speed until convergence is obtained. As a representative case study, we apply this method for the simulation of seakeeping behavior of a cruise ship in extreme dead ship conditions. For the validation of the proposed methodology, the first- and second-order motions, i.e., heave, roll, and the drift velocity, respectively, as well as lateral accelerations of the vessel were investigated for two cases of severe beam seas further combined with a constant strong wind load. The results were compared against experimental model tests.