TSP will clean off the grease and grime, but if the melamine is worn or peeling, it's toast and time to replace.

I'd really strongly suggest reevaluating your plan. Placing anything so expensive and permanant as granite on top of melamine cabinets that aren't in the best condition is a waste of resources.

Is this white melamine interior or exterior? I agree that a white melamine exterior that needs freshening might be past its prime and a good candidate for replacement before new counters are put in. Since you were talking about paneled cabinet doors, however, I don't know if you mean there's a mix of melamine and wood, or if the melamine is on the inside, which is also commonly done.

If it's in good shape and just gunky, you should be able to do wonders with mild dish soap (liquid for washing by hand, never dishwasher detergent). Vacuum first. Try a ribbed microfibre cloth with your suds; it will really get into the texture if there is one. Wet the cloth or sponge, not the cabinet. Don't use too much water. Rinse with a damp, clean cloth or sponge. Wipe dry with a rag or towel.

Melamine, being plastic, doesn't like chemical cleaners. I've heard that there's special stain remover for discolored melamine, but haven't seen it let alone used it. They might have it at a hardware store.

If the melamine is delaminated in places, you might be able to repair it with a curved syringe and the right kind of glue.

I use clorox cleanup on my white formica cabinets and counters. Works every time!