A novel technique for tuning the strength of quantum confinement in site-controlled semiconductor quantum dots (QDs) is introduced and investigated theoretically and experimentally. The method makes use of controlled local growth rates during metalorganic vapor phase epitaxy on patterned arrays of inverted pyramids. A model accounting for precursor migration and adatom incorporation predicts the tuning in QD thickness as a function of the pattern parameters. The results are in good agreement with experimental findings. This technique offers means for designing QD photonic structures with potential applications in QD-based cavity quantum electrodynamics and quantum information processing.