On-off kinetics observed in fluorescence intensity traces of single organic dye molecules have been assigned to molecules being either continuously emitting photons (on state) or occupying the triplet state (off state). To date this was mainly substantiated by the oxygen dependence of this behavior. By means of various experimental approaches, we show that indeed the occupation of the triplet state principally causes the off periods in some cyanine, dyes structurally related to frequently used DiI or Cy3. Comparing recorded traces for various cyanine dyes bearing different atoms at the 1- and 1'-positions, we demonstrate for the first time the consequences of the internal heavy atom effect on the on-off dynamics of single organic dye molecules. An additional pathway to populate the triplet state is proposed in order to explain the enhanced intersystem crossing quantum yield for the transition S, to T, for these molecules, which demand less free volume for rotation.