The free-space near-field diffraction of a transmissive periodic structure is studied with Fresnel-Kirchhoff diffraction formula. A new effect named Talbot spectra is proposed theoretically from the results, showing that under some conditions the spectra of an incident coherent polychromatic light can be completely restored after the diffraction. It can be contrasted with its monochromatic counterpart, Talbot images, which rebuild the images of a periodic structure at specific places in the near field. Taking a sinusoidal amplitude grating as an example for illustrating the idea, the general conditions and specific locations to find the restored spectra are obtained. The spectral switches involved are also presented.