Numerical calculation of minimum-energy paths and activation energy barriers for various atomic diffusion processes on fcc metal surfaces are presented. The computational method employed is the action-derived molecular dynamics that searches the approximate Newtonian trajectory on potential-energy surfaces. The minimization of a modified action, which facilitates the conservation of total energy and the control of kinetic energy, enables us to find efficiently the minimum-energy paths of complex microscopic processes. Diverse diffusion mechanisms on flat fcc substrates are investigated in this first part of the series. More complicated systems including surface steps are simulated in paper II.