The graphene layers are held by weak van der Waals' force, this should be comparable to the force holding solvent particles. Why doesn't graphite dissolve in organic solvent and separate into graphene layers?
While you cannot directly make a suspension or dispersion of graphene nano-platelets from graphite, it is possible to perform very similar process, called exfoliation, on graphite oxide, as graphite oxide has functional groups decorating it that reduce the functional strength of the van der Waal's forces between layers.
The exfoliation process can create a dispersion of graphene nanoplatelets in an organic environment, however, outside energy is often (if not always) necessary to promote the exfoliation of the individual layers. Most often, this energy is provided by sonicating the solution.
A very quick search of suggests that this method has been shown to work using NMP, DMF, THF, and ethelyne glycol. (see Paredes, Villar-Rodil, Martínez-Alonso, and Tascón, 2008 for more info)
Because lots of weak interactions lead to large attractive forces
The reason why graphite (or graphene) won't dissolve in common solvents isn't to do with the nature of the chemical forces holding the layers together: it has more to do with the sheer number of such weak forces in each small flake of the substance.
Each hexagon has only dispersion forces to bind it into a solid crystal. Compare to benzene which has just a single carbon hexagon. Those forces between hexagons mean that benzene crystallises at around 6 C. Disrupt them even slightly (by adding a methyl group to give toluene) and the solid doesn't form until you have cooled the substance about 100 degrees. The forces are weak, but notable in their effect. Benzene is soluble or miscible with many organic solvents because you only have to disrupt single benzene-benzene interactions to get a solution.
So why are layers of graphite not soluble? The same forces that makes benzene crystallize at such a high temperature are present but, in each tiny flake, there are millions of hexagons in the same plane. When even a microscopically tine such flake interacts with another flake the total force between them will be millions of times stronger than the forces between benzene molecules. That is a lot of attractive force to overcome if you want to get a flake into solution. You can't overcome the hexagon-hexagon interactions one by one as you do for benzene, you have to disrupt the whole sheet all at once and that is monumentally harder than doing one hexagon at a time.