We study the group-velocity-dispersion properties of a novel type of Bragg fibers. These new structures are cylindrically symmetric microstructured fibers having a high-index core (silica in our case), like in conventional photonic crystal fibers, surrounded by a multilayered cladding, which is formed by a set of alternating layers of silica and a lower refractive-index dielectric. The combination of the unusual geometric dispersion behavior shown by the multilayered structure and the material dispersion corresponding to the silica core allows us to design nearly constant chromatic dispersion profiles. In this work we focus our attention on flattened dispersion fibers in the 0.8 microm wavelength window and even on ultraflattened dispersion structures about the 1.55 microm point. We include configurations owning positive, negative, and nearly-zero dispersion in both wavelength ranges.