Nanoelectromechanical systems constitute a class of devices lying at the interface between fundamental research and technological applications. Realizing nanoelectromechanical devices based on novel materials such as graphene allows studying their mechanical and electromechanical characteristics at the nanoscale and addressing fundamental questions such as electron-phonon interaction and bandgap engineering. In this work, we realize electromechanical devices using single and bilayer graphene and probe the interplay between their mechanical and electrical properties. We show that the deflection of monolayer graphene nanoribbons results in a linear increase in their electrical resistance. Surprisingly, we observe oscillations in the electromechanical response of bilayer graphene. The proposed theoretical model suggests that these oscillations arise from quantum mechanical interference in the transition region induced by sliding of individual graphene layers with respect to each other. Our work shows that bilayer graphene conceals unexpectedly rich and novel physics with promising potential in applications based on nanoelectromechanical systems.