The growth of many crystalline materials occurs through lateral propagation of steps over the surface(1-3). A stepped texture is also characteristic of diamond grown by chemical vapour deposition (CVDM)(4-10). Diffusion of atoms on the surface is usually held responsible for the stepped growth of metals, but it has been thought(11-14) that the stronger bonding of adatoms should prevent this mechanism from operating in the case of diamond. Recent experiments(15) have, however, indicated that surface diffusion can take place during diamond growth. We have recently shown theoretically(16) that bridging methylene (CH2) groups on the {100} plane of diamond growing in the presence of hydrogen can migrate in a manner equivalent to surface diffusion. Here we show that this theoretical picture can be developed into an atomistic model that accounts for stepped growth of diamond. The stepped pattern can be understood in terms of the formation of surface-bound species from the gaseous precursors, followed by their migration by means of a series of surface chemical reactions involving covalent bond breaking and formation.