Aims:We analysed the consecutive height series of Hα spectra in solar limb spicules taken on the 53 cm coronagraph of Abastumani Astrophysical Observatory at the heights of 3800-8700 km above the photosphere. Our aim is to observe oscillatory phenomena in spicules and consequently to trace wave propagations through the chromosphere. Methods: We use the Discrete Fourier Transform analysis of Hα Doppler shift time series constructed from the observed spectra at each height. Results: Doppler velocities of solar limb spicules show oscillations with periods of 20-55 and 75-110 s. There is also clear evidence of 3-min oscillations at the observed heights. Conclusions: The oscillations can be caused by wave propagations in thin magnetic flux tubes anchored in the photosphere. We suggest the granulation as a possible source of the wave excitation. Observed waves can be used as a tool for spicule seismology, and the magnetic field strength in spicules at the height of 6000 km above the photosphere is estimated as 12{-}15 G.