I review domain coarsening phenomena in the non-equilibrium dynamics of discrete spin models with Z_2 symmetry. Standard classifications distinguish between "curvature-driven" processes, in which the interface motion is driven by surface tension (e.g. Glauber dynamics), and purely diffusive processes driven by multiplicative noise at the interfaces (e.g. Voter model). There are evidences that slight changes of the microscopic Voter-like rule (e.g. adding intermediate states) leads to the emergence of an effective surface tension. We explain this phenomenon deriving a macroscopic Langevin equation for these processes and studying it with field-theoretic and numerical methods.