The structure of the Cu(110) surface is studied at high temperatures using a combination of lattice-gas Monte Carlo and molecular-dynamics methods with identical many-atom interactions derived from the effective-medium theory. The anisotropic six-vertex model is used in the interpretation of the lattice-gas results. We find a clear roughening transition around T(R) = 1000 K, and T(R)/T(M) = 0.81. Molecular dynamics reveals the clustering of surface defects as the atomistic mechanism of the transition and allows us to estimate characteristic time scales. For the system of size 50 x 50, the time scale of the local roughening at 1150 K of an initially smooth surface is of the order of 100 ps.