We investigate the null controllability of the wave equation with a Kelvin-Voigt damping on the two-dimensional torus T-2. We consider a distributed control supported in a moving domain omega(t) with a uniform motion at a constant velocity c = (1, zeta). The results we obtain depend strongly on the topological features of the geodesics of T-2 with constant velocity c. When zeta is an element of Q, writing = zeta = p/q with p, q relatively prime, we prove that the null controllability holds if roughly the diameter of omega(0) is larger than 1/p and if the control time is larger than q. We also prove that for almost every zeta is an element of R+ \ Q, and also for some particular values including, e.g., zeta = e, the null controllability holds for any choice of omega(0) and for a sufficiently large control time. The proofs rely on a delicate construction of the weight function in a Carleman estimate which gets rid of a topological assumption on the control region often encountered in the literature. Diophantine approximations are also needed when zeta is irrational.