We present a new concept of fluorescence suppression in Raman microscopy based on micro-spatially offset Raman spectroscopy which is applicable to thin stratified turbid (diffusely scattering) matrices permitting the retrieval of the Raman signals of sublayers below intensely fluorescing turbid over-layers. The method is demonstrated to yield good quality Raman spectra with dramatically suppressed fluorescence backgrounds enabling the retrieval of Raman sublayer signals even in situations where conventional Raman microscopy spectra are fully overwhelmed by intense fluorescence. The concept performance was studied theoretically using Monte Carlo simulations indicating the potential of up to an order or two of magnitude suppression of overlayer fluorescence backgrounds relative to the Raman sublayer signals. The technique applicability was conceptually demonstrated on layered samples involving paints, polymers and stones yielding fluorescence suppression factors between 12 to above 430. The technique has potential applications in a number of analytical areas including cultural heritage, archaeology, polymers, food, pharmaceutical, biological, biomedical, forensics and catalytic sciences and quality control in manufacture.