No loose articles are allowed on rides... you would have to put them away in backpack/fanny pack/purse, etc on rides that allow carry ons.. seeing as they aren't metal you could put them in your pocket for locker only rides, but that doesn't seem like something would want to do, so leave in locker with everything else.
Typically if I wear one, I have an insulated backpack since I typically need cold water and I put my towel in there to keep cool with the cold water / ice while I wait in line (since typically it's air conditioned so there would be no need for it).
However, I generally have it not only lying around my neck area, as well as tucked under my shirt / collar area, so not only it stays on you that way, but also more of your skin is exposed to the towel and keeps you cool. So, I would imagine doing that should be fine as it shouldn't go anywhere (especially rides where the bar goes over your shoulders). If they notice it and it's not good enough,, as others have said, tie it around (if possible), or put it in your pocket perhaps.