Should I be concerned about my periods being really heavy?

Yes. One persons "really heavy" flow is another's average flow. It is hard to quantitate what someone means by heavy flow but if it is a change from your normal or usual flow you should be concerned. Heavy flow can lead to significant anemia which can have other consequences. It can usually be treated to avoid the problem and help you save money from buying a lot of pads or tampons. See a gynecologist!
Anemia. If your getting worn down, anemic or just frustrated, then sure, but otherwise its ok to tolerate them.