Break out the ladder and take a look at the damage. Put on your gardening gloves and see what type of debris you're dealing with. This is also a good time to check for any damage or wear like cracks and leaks in the downspouts.
Instead of using your hands to remove the debris, fashion a homemade gutter scoop from an empty, plastic milk jug or soda bottle. Cutting the corners in the shape of a V will provide a shape workable for gathering the most debris. You can also use kitchen utensils, like spatulas and old serving spoons, for narrow or hard-to-reach areas.
Instead of throwing all the gutter debris onto your yard or aiming for a small buket, use a camping tarp or old bed sheet as a collection area. Set up the tarp or sheet in a large square or rectangle, and aim your scoopfuls of debris at the shape. When you're finished, you can simply fold up the tarp or sheet and dump the debris in a yard waste bin, compost pile or leaf bag.
When you're finished scooping out the debris, you can flush out all the remnants from the gutters and downspout with your garden hose.