[en] The observations of the pseudo-periodical chessboard (CB) microstructure in metal and ceramic solid solutions indicate that this is a general phenomenon. We propose a theory and three-dimensional (3-D) computational modeling explaining the origin of the CB microstructure in the cubic → tetragonal decomposition. The 3-D modeling demonstrates that the formation of two-phase CB structures is contingent on the formation of a compositionally stabilized precursor state with the tweed structure that is spontaneously formed at the initial stage of the transformation. The modeling has shown that this tweed structure is a distribution of spatially correlated tetragonal nanodomains whose spatial arrangement has the CB topological features. This precursor tweed structure serves as a template for the precipitation of the equilibrium cubic phase. The CB-like tweed template channels the microstructure evolution towards the two-phase CB structure whose complex and detailed 3-D geometry is in excellent agreement with electron microscopic observations. The thermodynamic analysis and obtained evolution sequences allow us to formulate the necessary thermodynamic, structural and kinetic conditions for the CB structure formation. Reasons for its relative stability are discussed. It is also shown that the coherency between the cubic and tetragonal phases comprising the CB structure produces the stress-induced tetragonality of the cubic phase, orthorhombicity of the tetragonal phase, and rotations of cubic phase rods. These effects should diminish and disappear upon lifting of coherency