Dye aggregates and their excitonic properties are of interest for their applications to organic photovoltaics, non-linear optics, and quantum information systems. DNA scaffolding has been shown to be effective at promoting the aggregation of dyes in a controllable manner. Specifically, isolated DNA Holliday junctions have been used to achieve strongly coupled cyanine dye dimers. However, the structural properties of the dimers and the DNA, as well as the role of Holliday junction isomerization are not fully understood. To study the dynamics of cyanine dimers in DNA, molecular dynamics simulations were carried out for adjacent and transverse dimers attached to Holliday junctions in two different isomers. It was found that dyes attached to adjacent strands in the junction exhibit stronger dye-DNA interactions and larger inter-dye separations compared to transversely attached dimers, as well as end-to-end arrangements. Transverse dimers exhibit lower inter-dye separations and more stacked configurations. Furthermore, differences in Holliday junction isomer are analyzed and compared to dye orientations. For transverse dyes exhibiting the smaller inter-dye separations, excitonic couplings were calculated and shown to be in agreement with experiment. Our results suggested that dye attachment locations on DNA Holliday junctions affect dye-DNA interactions, dye dynamics, and resultant dye orientations which can guide the design of DNA-templated cyanine dimers with desired properties.