This paper highlights the interest of shunt-diode rectifiers to optimize the performances of piezoelectric vibration energy harvesters based on linear load adaptation. It provides, for the first time, a full analytical study of the input power and frequency bandwidth of such a system. We show that this architecture is well-suited for piezoelectric energy harvesting. In the presented approach, a so-called shunt diode is connected across the electrodes of the piezoelectric transducer, which makes the piezoelectric voltage unidirectional instead of alternating. Thus, DC-DC converters usually used for power optimization can be directly connected to the transducer. We show that weakly-coupled linear inertial piezoelectric energy harvesters generate higher power when connected to a shunt-diode rectifier than to a bridge rectifier. Moreover, the frequency bandwidth is enlarged by the use of the single-diode shunt rectifier, especially for strongly-coupled piezoelectric generators. The proposed architecture is also interesting for low piezoelectric voltages, at which voltage drops across the diodes used for AC-DC conversion critically degrade the power efficiency. Experimental results validate the presented analytical study and highlight the higher efficiency of the shunt-diode rectifier circuit. Compared to a standard bridge rectifier architecture, we demonstrate an increase of +43% output power on a weakly-coupled generator and an increase of +5% bandwidth on a strongly-coupled generator.