The quantum resonances (QRs) of the kicked particle are studied in a most general framework by also considering arbitrary periodic kicking potentials. It is shown that QR can arise, in general, for any rational value of the Bloch quasimomentum. This is illustrated in the case of the main QRs for arbitrary potentials. In this case, which is shown to be precisely described by the linear kicked rotor, exact formulas are derived for the diffusion coefficients determining the asymptotic evolution of the average kinetic energy of either an incoherent mixture of plane waves or a general wave packet. The momentum probability distribution is exactly calculated and studied for a two-harmonic potential. It clearly exhibits additional resonant values of the quasimomentum and it is robust under small deviations from QR.