A route toward lineshape engineering of Fano resonances in photonic structures is theoretically proposed, which uses dynamic modulation of the refractive index of a microcavity. The method is exemplified by considering coupled-resonator optical waveguide systems. An exact Floquet analysis, based on coupled-mode theory, is presented. Two distinct kinds of resonances can be dynamically created, depending on whether the static structure sustains a localized mode or not. In the former case a single Fano resonance arises, which can be tuned in both frequency and line width by varying the refractive index modulation amplitude and frequency. In the latter case two resonances, in the form of narrow asymmetric dips in the transmittance, are found, which can overlap resulting in an electromagnetically-induced transparency effect.