The origin of frequency gaps in the dispersion relation of periodic, quasi-periodic, and random photonic structures consisting of different arrangements of dielectric cylinders has been investigated. For TM polarization it was found that the formation and properties of gaps are strongly affected by Mie resonances of a single cylinder. Both the spectral position and size depend on the properties of this single scatterer. In contrast, for TE polarization no correlation between the scattering properties and bandgap formation was found, as Mie resonances are spectrally not well separated. For the inverted structure consisting of air cylinders in a dielectric material, the frequency gaps depend on the spatial arrangement of the cylinders because no pronounced Mie resonances exist in this case.