We describe single-electron tunneling through molecular structures under the influence of nanomechanical excitations. We develop a full quantum-mechanical model, which includes charging effects and dissipation, and apply it to the vibrating C-60 single-electron transistor experiment by Park et al. ( Nature, 407 (2000) 57). We find good agreement and argue vibrations to be essential to molecular electronic systems. We propose a mechanism to realize negative differential conductance using local bosonic excitations.