Here's the thing: it's a little silly to be looking for a "natural" method for something that's about thwarting nature. Essentially, NO methods of birth control are "natural," and certainly not hormonal contraceptives. And just because they aren't natural doesn't make them weird or problematic.
It definately sounds like hormonal methods aren't for you (though it should be mentioned that lately many women who have had troubles with the pill or depo have been reporting the nuvaring is working for them). And no, I'd not suggest an IUD for younger women, pre-childbirth, and neither would most GYNs who serve young women, in my experience.
That leaves you with cervical caps or diaphragms (which if you use a Keeper or tampons, are really no different), as well as condom use. If you want to combine something more "natural" with condoms for backup, you can start charting your cycles as well, and then abstain from ALL procreative sexual activity when you discover you're most fertile, using condoms all other times.