We study the kinetics of growing cell populations by means of a kinetic Monte Carlo method. By applying the same growth mechanism to a two-dimensional (2D) and a three-dimensional (3D) model, and making direct comparison with experimental studies, we show that both models exhibit similar behavior. Based on this we propose a method for establishment of a mapping between the 2D and 3D results. Additionally, we present an analytic approach to obtain the time evolution, and show in case of the 3D model how synchronization effects can influence the growth kinetics. Finally, we compare the results of our models to experimental data of the growth kinetics of 2D monolayers and 3D NIH3T3 xenografts in mice.