A Raman spectrometer technique is described that aims at suppressing the fluorescence background typical of Raman spectra. The sample is excited with a high power (65W), short (300ps) laser pulse and the time position of each of the Raman scattered photons with respect to the excitation is measured with a CMOS SPAD detector and an accurate time-to-digital converter at each spectral point. It is shown by means of measurements performed on an olive oil sample that the fluorescence background can be greatly suppressed if the sample response is recorded only for photons coinciding with the laser pulse. A further correction in the residual fluorescence baseline can be achieved using the measured fluorescence tails at each of the spectral points.