Welcome aboard, JB.
Nothing beats a good, old-fashioned bath using soap and water, but that does require complete disassembly.
For cleaning without disassembly, I have used Orange Glo on sticky, nicotine-type stains and had really good results. It helps remove grease, wax, and accumulated dirt. Yes, Orange Glo is technically for wood floors, but it works amazingly well on plastic, and so far, leaves no damage. Just be sure to not spray it on heavily - just a light mist is enough to get it done.
You can find Orange Glo pretty much everywhere - Walmart, Target, K-mart, Home Depot, etc.