Micron-sized latex particles with single-stranded DNA grafted to the surface have been used as a model system to study DNA mediated interactions. A new approach of tuning the interactions between particles is proposed, which allows for a gradual change of the assembly rate for fixed physical conditions of a solution by combining hybridizing "linker" DNA with nonhybridizing "neutral" DNA. The effect of linker/neutral DNA ratios on particle assembly kinetics and aggregate morphology has been experimentally investigated for a range of ionic strengths. The conditions for controlling various assembly morphologies have been identified, and the involved attractive and repulsive interactions have been described and explained for the proposed approach. The calculated attractive-repulsive behavior is in good agreement with experimental results. The described approach provides general perspectives for further fine-tuning DNA-mediated assembly systems.