Cut up a pair of pantyhose so that you have two "legs." Fill the legs with calcium chloride.
Lay the sausage-shaped hose packets perpendicular to your gutters over the roof. You should position them at least 18 inches apart for the best results.
Chisel a pathway through the ice so that any runoff from the roof will have some place to drain. You don't have to chisel the entire dam away, as the water which runs off will be above freezing and will help melt the rest.
Attach a garden hose to a spigot and run it over the downspout. Turn the water on to get a small stream. Place the garden hose over the downspout so the water drips into the spout. As long as the water is warmer than freezing, it will help break up the ice.
Install roof-mounted heating cables in the gutters once they have been cleared to keep the ice from forming again.