Your period when you're on the pill isn't a real period. You haven't ovulated (or you shouldn't have anyway), so there is no uterine lining to get rid of. The pill basically works by tricking the body into thinking you are pregnant. The bleed you get on the pill is your body going into withdrawal due to the sudden drop in synthetic hormones, caused by switching from the pill to the sugar pills in the packet. It's not the same thing as your period, and it's not necessary.
The man who created the pill (or who was the main driving force anyway, of course many people worked on it), was a staunch Catholic. He very much wanted the church to accept the pill, so he created something that could also be said to help married women regulate their cycles, in order that they find it easier to get pregnant. To be able to claim this, women needed to get their 'period' so they could start charting it and figure out their cycle. There's no reason to have a bleed when you're taking the pill, other than one man's desire to gain the Catholic church's approval. Of course the church condemned it anyway (at the time).
Skipping these 'periods' is actually a good thing. If the pill had been made solely with women's health in mind there would be no sugar pills, and you wouldn't have a bleed. The body isn't used to having so many bleeds, women used to be either pregnant or nursing for much of their adult lives, and thus had far fewer periods. Suddenly having on average twice as many over the course of a lifetime takes it's toll. Just being on the pill in the first place and having these 'fake' bleeds helps with that a bit, but skipping over it altogether at least some of the time is the ideal.
In the UK a lot of doctors prefer you have a bleed every 3 months or so, in the US it's more like 6 months, in most places though doctors don't mind if you skip it for years. I skipped it for 6 years in my 20's, and I think it's been about 3 years since I've had a bleed now.
The only downsides are that you may get a little breakthrough bleeding anyway while you're body adjusts, and it's slightly harder to tell if you're pregnant, so you should be peeing on a stick every month or so just to be sure. The pregnancy issue is the main reason some doctors prefer you have a bleed now and then, because of course there are some people who won't take a test regularly or notice the signs of pregnancy until they're well into it. But if you are capable of being responsible with the pill and you can remember to take a test and be on the lookout for symptoms, there's no issue.