Rolling element bearings are among the most important components of rotating machinery, being the interface between the stationary and the rotating parts. Bearing failures might cause accidents and unexpected machine breakdown. Therefore, there is an important need for novel condition monitoring techniques focusing towards early detection and accurate identification of emerging faults. The key aim of this paper is the proposal of the concept of Multiband Demodulation Analysis (MDA), performed using a bank of bandpass Gabor filters combined with the Teager-Kaiser Energy Separation Algorithm (ESA) for condition monitoring of rolling element bearings. The demodulation takes place in multiple frequency bands, following a multiband analysis scheme, in order to isolate the strongest modulation components in each band. Finally, a maximum average energy tracking process over the various frequency bands is used to yield short-time measurements of the multiband signal modulation energy and the demodulated instant amplitude and frequency. The method is applied, tested and evaluated on vibration signals, captured on two test rigs and its performance is compared with a state of the art diagnostic approaches, the Kurtogram and the empirical mode decomposition, achieving promising results.