Propagation of longitudinal acoustic waves in a one-dimensional piezoelectric structure with space-time modulated electrical boundary conditions is investigated. An analytical model allowing the calculation of eigenmodes for spatially continuous shifts of electrical boundary conditions is compared with finite difference time domain simulation results for a discrete set of time-varying spatially fixed conditions. Both models predict that such a structure behaves as a nonreciprocal device exhibiting unidirectional propagation properties in some frequency ranges. The modulus and direction of the modulation speed vector strongly affect this nonreciprocal behavior. Moreover, other nonlinear phenomena such as frequency up and down conversions, wave packet distortion, and parasitic echoes occurring in such systems are investigated in detail. The importance of these phenomena is discussed in the context of nonlinear acoustic wave-based components for radio-communication systems.