Condoms are without a doubt the most popular form of non-hormonal birth control around. Their biggest advantage is that they also severely reduce the risk of being infected with sexually transmitted diseases including HIV. Another great benefit of condoms is that their use can cease without the involvement of a medical professional (unlike the copper coil, for instance). Many men are worried that condoms will reduce their sexual pleasure, but they should at least take into account the alternatives... taking a pill every day is also a hassle for women. Men also have responsibilities!
Although the hormone-emitting Mirena IUD is widely used now, non-hormonal coils are still available and still effective. The Paraguard is a good example of a T-shaped copper intrauterine device that doesn't include hormones. Intrauterine devices are very effective, can remain in place for many years, and do not have many side effects. They do require an appointment to remove them.
The diaphragm may be an "ancient" pregnancy prevention device, but it is actually still in use! Modern diaphragms are made of silicone and are relatively inexpensive birth control devices. They don't have to "spoil the moment" like condoms, because they can be inserted hours before you actually intend to use them. Though, as someone who's never seen or used a diaphragm, it sounds a little horrid (sorry). Spermicides can be inserted into diaphragms to make them more effective.
Permanent birth control methods are hormone free, and a good option for many people who are sure that they never want to have a biological child (again) anytime in the future. These permanent measures can come in different forms, including a vasectomy for men, and a tubal ligation (which again comes in different varieties) for women. But did you know there are also less medically invasive permanent contraceptives around now, that do the job just fine? They are called Adiana or Essure permanent birth control.