We theoretically propose a scheme to achieve multi-sideband phonon blockade in a modulated circuit quantum acoustodynamics system, where a high-frequency bulk acoustic-wave resonator is piezoelectrically coupled to a transmon qubit. By periodically modulating the qubit frequency, we theoretically demonstrate a Floquet-engineered Jaynes-Cummings interaction that can extend the conventional blockade condition from single-phonon resonance to multi-sideband resonance in the strong-coupling regime. Under weak driving conditions, we first show the phonon blockade at the $0$-th order sideband due to the strong anharmonicity of the dressed-state energy levels, and characterize the blockade performance by combining the second-order correlation function $g^{(2)}(0)$ and the single-phonon population $P_1$. Crucially, it is revealed that the detuning between the phonon mode and the qubit can lead to a trade-off between $g^{(2)}(0)$ and $P_1$ by altering the dressed-state energy-level structure and composition, and thus can be used to manipulate the purity and brightness of the single-phonon source. Then we extend the phonon blockade to multiple tunable Floquet sidebands, and analyze the influence of the system parameters on the phonon blockade. The proposed scheme enables the manipulation of the phonon blockade at multiple distinct frequencies without modifying the intrinsic system parameters, which reduces the experimental complexity and improves the operational flexibility, thereby facilitating the potential applications of circuit quantum acoustodynamics systems in emerging quantum technologies.