Because you have to pick your battles at this age, I am going to give you permission to let their rooms be messier than you would like. I have no authority whatsoever to do this, but I have found that giving teens a little autonomy over things that don't matter much, goes a long way. This will enable you to have "capital" when there are things you really need to control in their lives.
However, if you let it go for too long, it could be a health hazard. In our family we make their Saturday job to clean their bedrooms (and sometimes it happens every other Saturday, depending on soccer game schedules). They have to do this before they go out with their friends. By this time, they've all been taught how to clean the bathrooms, and various other Saturday chores, so I'm not worried about training them for life, so I don't mind giving them their rooms each week instead of other things. If you do this, then it will at least get cleaned once a week and you won't have to worry about mice or other such things.
I know, it sound less clean than you would like, especially when you go under their bed and find that they have a toilet paper collection, but I promise, it will pay off. If you pick your battles your relationship will be stronger. And that's the most important thing at this rocky stage.