I usually take the pill for 3 months straight and then stop, have my period and resume for another 3 months. Can I take it for 6 months straight and not have any problems?

I believe 6 months is the max. As someone who doesn't naturally have a period often and am on the pill for that reason, my doctor said it's not an issue until you miss your period for 6 months. Your peroid is there for health reasons, not simply to annoy you, which is why even the pill designed so you only have a period 3 times a year, you still need to have a period.

I would honestly talk to your doctor. My pharmacist won't even give me my pills if I need them sooner than she thinks it required based on them being designed for a period every month.

Read the insert pamphlet that comes with your prescription.

You know, the only reason they have woman take the placebo for a week every month is that it makes some women feel better to still have a period. They like to be on a cycle and have a confirmation once a month that they are not pregnant. And on birth control it isn't even a real period, it's what they call withdrawal bleeding. The hormonal changes don't simulate a real period at all.
Many women have taken regular birth control pills in three month intervals for many years before they came out with Seasonale (the one that the prescription is for three months straight). Now they just came out with Lybrel, you never take a placebo all year. The daily pill has the same hormones in the same amounts as other pills you take for one month. The only added side effect is a higher chance of breakthrough bleeding (when you take a placebo for a week each month you are just able to predict the bleeding). But many women don’t get breakthrough bleeding at all. I took the birth control shot Depo-Provera for 5 years (you are not supposed to have a period at all while taking it) and I never had breakthrough bleeding.
Now I do the pill, but I researched the crap out of it first. There are over 40 different brands of birth control pill that all work but many have slightly different amounts of ethinyl estadiol (the estrogen hormone) and different types and amounts of the progesterone hormone (there are 9 different types!). Depending on the side effects you are prone to, you get the type that is the best for combating that. I take a pill that is made for three weeks on/one week off, but take it for three months straight. It has exactly the amounts and types of hormones I need to have no side effects. The problem with Seasonale or Lybrel is there is only one amount of the hormones in them and that might not be the best one for you. And they are name brand, I take generic and it’s less than half the price! I could probably take it for 6 months or a whole year, since I know breakthrough bleeding is not a problem for me, but I don’t mind a period once every three months.

'Serazet' my wife was given this pill to take continuously with no breaks and no periods, this is in the UK I dont know if you have it in the US or if it has a different name, your Doc can advise you better if you want to change to it :o)

That's really bad, the schedule is designed the way it is for a reason, by doing this you are seriously compromising your health, although you may not feel adverse effects now, you are greatly increasing your risk of breast cancer and infertility by messing with your hormones