There are the FAM methods and other fertility awareness, but it really is a science, and many people end up getting pregnant this way- because it does really require you being very intimate with yourself with knowing how your body works- checking out secretions, monitoring your temperature there, etc... plus, not everyone has the same ovulation pattern, so there is always the potential for pregnancy.
I have been looking into it too, and I had been using a diaphram, which is non-hormonal, but, well, not good for juggling a small child and free time to even get to have sex. Again, it requires a bit of intimate knowledge on your part of yourself to get it in place, and know when it is in place. In addition, I got pregnant even with the diaphram.
So, this time round, we are going for the copper iud. The worst I have heard about it is that for some people it causes cramps, but I had cramps anyway... so I'm not that concerned. It lasts up to ten years, but you can take it out at any point before that, easily.
My understanding is that the mirena, has some hormones, but it is localized hormone release, so it does not go through your entire system the way oral birth control does.