Let $P$ be a collection of $n$ points in the plane, each moving along some straight line at unit speed. We obtain an almost tight upper bound of $O(n^{2+ε})$, for any $ε>0$, on the maximum number of discrete changes that the Delaunay triangulation $\mathbb{DT}(P)$ of $P$ experiences during this motion. Our analysis is cast in a purely topological setting, where we only assume that (i) any four points can be co-circular at most three times, and (ii) no triple of points can be collinear more than twice; these assumptions hold for unit speed motions.