Resonance-induced enhancement of fluorescence from a thin dye layer coated on a 9.8-\ensuremath{\mu}m-diam glass fiber has been observed. Enhancement occurs when the wavelengths of incident radiation and/or inelastic emission are resonant with natural modes of oscillation of a circular dielectric cylinder. These dielectric resonances are comparable to surface plasmon resonances in metallic microstructures, which are partially responsible for the surface-enhanced Raman effect.