We present a numerical method to plot the root locus of Single-Input-Single-Output (SISO) dead-time systems with respect to the controller gain or the system delay. We compute the trajectories of characteristic roots of the closed-loop system on a prescribed complex right half-plane. We calculate the starting, branch and boundary crossing roots of root-locus branches inside the region. We compute the root locus of each characteristic root based on a predictor-corrector type continuation method. To avoid the high sensitivity of roots with respect to the locus parameter in the neighborhood of branch points, the continuation method relies on a natural parameterization of the root-locus trajectory in terms of a distance in the (characteristic root, locus parameter)-space. The method is numerically stable for high order SISO dead-time systems.