There is a wide variation of normal when you talk about 'heavy periods'.
In a normal menstrual cycle (usually lasts 3-8 days) the flow will vary depending on where you are in your period. Some women have pink, brown or light red staining before their period establishes a true flow. This staining may last a few hours, or it may last a few days, and then the heavier bleeding will begin.
It is not uncommon to have cramps with the period (dysmenorrhea), and it is common to even have clots form during times of heavy flow. Clots in their own right are not abnormal, as long as they are not excessive. Clots may seem more significant after waking from sleep, or standing after prolonged sitting because blood comes out of the uterus and collects in the vagina and congeals into a clot. When you stand up, the clot comes out.
If you soak more than a pad or a tampon in a half hour to an hour on a regular basis, this may be too much, and you should speak to your doctor. Again, changes from your baseline amount of period blood should also be looked into. If you are feeling weak, dizzy or overly exhausted during or after a period you should have blood drawn to make sure you are not anemic (a drop in your blood count because of blood loss). If you do have these symptoms and/or you do become anemic, your doctor will need to evaluate you right away.