At the hotel I worked at in college, for both lawn chairs and the floating stuff we had for the customers:
Spray 409 on the mats fairly heavy.
Let this sit for few minutes.
Scrub brush the mats and toys.
RINSE, RINSE, RINSE.
All of the suntan oils, lotions, etc came off - the white plastic was always white. We did this like once or twice a week for a deep clean (depending on use) and in-between we'd spray the stuff down with disinfectant from housekeeping (who knows what was in that stuff! I would let it sit for the time on the can and then rinse it off per the instructions!)
You might also use diluted "Simple Green." It usually worked on the stuff; however, I always felt that it took longer to work and required more scrubbing than the 409.
-WC