standard cleaning procedures in core analysis often involve a cleaning step by Soxhlett extraction with a chloroform-methanol-water azeotrope (the water is usually already present in the core) in a reflux apparatus (see https://en.wikipedia.org/wiki/Soxhlet_extractor). In principle you can use this cleaning procedure also for cuttings.

However, when using oil based mud can be problematic for some of the later analysis of the cuttings because of the chemicals used to stabilize the oil-based mud (e.g. emulsifiers) can change the wetting properties permanently, i.e. is not cleaned by the above mentioned Soxhlett reflux. That is an issue for any two-phase flow property where the wettability state of the sample matters. But you can still get porosity, permeability, Hg-air capillary pressure (if the cuttings are large enough), see e.g. 

http://www.cydarex.fr/files/lenormand_SCA2010_A073.pdf