Lack of research. Long-term studies haven't been done to find out how these drugs will affect your body if you take them for several years. Both extended-cycle and continuous birth control drugs expose you to hormones for longer periods of time than normal birth control pills, which could affect your health. Some data suggests they could affect bone mass and breast health in the long run, especially among teens.
Pregnancy. Your period provides monthly proof that you aren't pregnant. If you do conceive while on these drugs, you might not realize it as quickly -- so you might not get prenatal care from your OB/GYN early enough.
Spotting. These drugs can cause spotting or bleeding between periods, sometimes heavy enough to require a pad. (This often lessens with time.) If you'd rather bleed on a set schedule and not worry that you might bleed or spot any day of the month, these drugs might not be for you.
Blood clots. Any form of birth control pill makes women more likely to get blood clots, especially those who smoke and are over 35.