The exploitation of bioactive natural sources to obtain new anticancer agents with novel modes of action may represent an innovative and successful strategy in the field of medicinal chemistry. Many natural products and their chemical analogues have been proposed as starting molecules to synthesise compounds with increased biological potential. In this work, the design, synthesis, and characterisation of a new series of N4,N4-dimethylated thiosemicarbazone Cu(II), Ni(II), and Pt(II) complexes are reported and investigated for their in vitro toxicological profile against a leukaemia cell line (U937). The antiproliferative activity was studied by MTS assay to determine the GI50 value for each compound after 24 h of treatment, while the genotoxic potential was investigated to determine if the complexes could cause DNA damage. In addition, the interaction between the synthesised molecules and DNA was explored by means of spectroscopic techniques, showing that for Pt and Ni derivatives a single mode of action can be postulated, while the Cu analogue behaves differently.