Cleaning Vinyl  

Start with something not-so-strong
like soap and water, then work your way up if necessary.
The progression suggested by the Naugahyde people is:
1. A solution of 10% household liquid dish soap with warm water applied with a soft damp cloth.
2. If that doesn't work, a solution of liquid cleanser and water applied with a soft bristle brush. Wipe away the residue with a water-dampened cloth.
3. If that doesn't work, dampen a soft white cloth with lighter fluid (naphtha) and rub gently. Rinse with a water-dampened cloth.
4. If that doesn't work, dampen a soft white cloth with a solution of household bleach (sodium hypochlorite); 10% bleach, 90% water. Rub gently. Rinse with a water-dampened cloth to remove bleach concentration.
5. If that doesn't work, allow a 1:10 diluted bleach solution to puddle on the affected area or apply with a soaked cloth for approximately 30 minutes. Rinse with a water-dampened cloth to remove any remaining bleach concentration.
http://www.naugahyde.com/cleaning.html

I used
one of the magic sponges and it worked wonders. Try this at your own risk though. Test an inconspicuous spot first.
Seems like many of the boat owners use it for their seats.

kartell plastic cleaner
I use the Kartell Plastic Cleaner on all of my synthetics including laminate and vinyl seats and it cleans wonderfully. I was skeptical at first but after I took the plunge and got some I wont be going back to another cleaner. An old but clean soft toothbrush in conjunction with the cleaner should get the job done.

Tips and Tricks
We have a Milo Baughman 1960's chair and a 1970s couch and chair set, both in vinyl, both through several sets of owners and abused. Both needed quite a bit of cleaning.
Two-step process:
1) Diluted dishwashing soap (like Palmolive or Dawn) and warm water and a slightly damp rag/toothbrush
2) Meguiar's Gold Class Leather Cleaner and Conditioner
They cleaned up pretty well - not perfectly, but much better than expected.

THANKS A MILLION!
To everything who replied.
I will try these things out and see how it goes.

Good quality vinyl is pretty durable stuff.
I would be leery of using any petroleum-based solvents or strong oxidizers, though, and would advocate the mild dishwashing detergent approach to start. That said, I've gotten great results using that "409" product and a toothbrush on a set of kitchen chairs I recovered twenty years ago. Just be sure and rinse well. Boaters and auto enthusiasts swear by 303 protectant for restoring faded color and as a UV inhibitor, but I find it tends to make vinyl a bit too shiny and slippery for my taste.