The distribution of spatial domain structures originated during one and three dimensional Poisson-Voronoi transformations are computed analytically extending the recently obtained re- sults for the two dimensional case. The presented method gives a full description of the developed microstructure and is valid for tessellations of any dimensionality. The temporal and spatial depen- dences of the domain structure are completely discriminated and separated, showing the existence of geometric configurations independent of time. A single computation of the probability distribution of these geometric configurations allows us to calculate the total free-boundary and size probabil- ity distributions at any desired time. The obtained results show full agreement with stochastic simulations and reproduce completely the previously existing partial results. A discussion about the potential applications of the method to the calculation of other geometrical properties and the characteristics of the final static structure leading to a gamma distribution of sizes is also presented.