HARVEST
You can begin to harvest sunflower seeds as soon as the center flowers turn brown or the backs of the heads turn yellow, to prevent birds from stealing them. Cut them, leaving a piece of stem to hang them in a well ventilated place to finish drying. Cover them with netting, paper sacks with holes or cheesecloth to catch falling seeds as they dry.

They can be allowed to dry on the stalk, but you'll have to cover them this way to keep the birds from eating them all before you can harvest them for yourself!

If you've grown sunflowers for the purpose of feeding birds, you can either leave them in the ground, or harvest the heads as above, then hang them in the yard or garden when they are ready. This method has an advantage in that you can dole out the heads over the winter, instead of seeing the seed all eaten within a few weeks.
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http://extension.missouri.edu/extensioninfonet/article.asp?id=3404