Let P be a collection of n points in the plane, each moving along some straight
line and at unit speed.
 We obtain an almost tight upper bound of O(n^(2+epsilon)), for any epsilon>0,
on the maximum number of discrete changes that the Delaunay triangulation 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.