Magnetic noise due to thermally activated movements of flux vortices has been calculated taking into account fluctuations modes of nonrigid vortices. It has been shown that at low frequencies, below the crossover frequency, the noise spectrum of a layered superconductor is identical to that of a continuous material. Three regimes of spectral behavior, lnω, ω-1/2, and ω-3/2, have been predicted to be present in this frequency range. Characteristic frequencies separating different regimes depend on the geometry of the flux pickup loop. At high frequencies, above the crossover frequency, bending of vortices leads to a Lorentzian shape of noise spectra. The value of the crossover frquency is not influenced by the particularities of the flux-measuring arrangement and depends only on the material properties and applied magnetic field.