Alfvénic turbulent cascade perpendicular and parallel to the background magnetic field is studied accounting for anisotropic dispersive effects and turbulent intermittency. The perpendicular dispersion and intermittency make the perpendicular-wave-number magnetic spectra steeper and speed up production of high ion cyclotron frequencies by the turbulent cascade. On the contrary, the parallel dispersion makes the spectra flatter and decelerate the frequency cascade above the ion cyclotron frequency. Competition of these factors results in spectral indices distributed in the interval [-2, -3], where -2 is the index of high-frequency space-filling turbulence and -3 is the index of low-frequency intermittent turbulence formed by tube-like fluctuations. Spectra of fully intermittent turbulence fill a narrower range of spectral indices [-7/3, -3], which almost coincides with the range of indexes measured in the solar wind. This suggests that the kinetic-scale turbulent spectra are mainly shaped by the dispersion and intermittency. A small mismatch with measured indexes of about 0.1 can be associated with damping effects not studied here. Key Points Dispersion and intermittent effects are included in an Alfvenic turbulence model Spectra of fully intermittent turbulence fill a narrower range of spectral indices [-7/3, -3] The kinetic Alfven wave can reach the ion cyclotron frequency through the turbulent cascade. © 2015 American Geophysical Union. All Rights Reserved.