Semiconductor quantum dot (QD) structures can be used as a model for understanding the effect of the microscopic structure, symmetry of crystals, and molecules on their macroscopic properties. In this work, the results of two theoretical approaches demonstrate that the spin dynamics in ordered QD structures depends on the size, spatial configuration, and topology of the object built of QDs. It was shown that the spin dynamics in QD structures with the hopping regime of conductivity significantly differs from the spin dynamics in two-dimensional (2D) and three-dimensional (3D) structures being at the other side of the metal-insulator transition. The special character of the effective magnetic field $\ensuremath{\delta}\mathbf{H}$ fluctuations appearing only during tunneling between quantum dots is responsible for the insensitivity of spin relaxation times to the magnitude of the external magnetic field in infinite QD structures (2D square lattice and 1D linear QD chain). In finite QD structures (QD rings and linear chains), an external magnetic field ${\mathbf{H}}_{0}$ is directly involved in the spin relaxation process and spin is lost due to interaction with a special combination of fields $\mathrm{\ensuremath{\Delta}}\mathbf{H}\ensuremath{\sim}[{\mathbf{H}}_{\mathbf{0}}\ifmmode\times\else\texttimes\fi{}\ensuremath{\delta}\mathbf{H}]/\ensuremath{\delta}H$ that leads to an unusual orientation dependence of ESR linewidth, recently observed for QD chains. It was shown that the ordering of QD structures can be used for the conservation of spin orientation. For 1D finite quantum dot chains, the ordering can provide the stabilization of all spin components ${S}_{x},{S}_{y}$, and ${S}_{z}$, while for ringlike molecules only ${S}_{z}$ polarization can be stabilized. The results obtained in this work can be useful for development of novel semiconductor devices and in quantum information processing.