Usually, people pick just the flowers.
And that is a waste of good herb: the stems of chamomile (Matricaria recutita) are perfectly useful, if weaker than the yellow flower.
It's easiest to just pull up a bunch of whole plants, roots and all - they're annuals which lean on one another, and their taproots are really pretty small.
If your plants are in full flower they're about ½ m tall (perhaps 1.5'). The lower half is mostly yellow leaf, if your bunch was pulled up in a lushly growing spot. Break the stems one by one where the leaf turns green, and spread the top bits on a bit of old bedsheet laid on top of a thick layer of newspapers.
Let dry in a shady airy spot for 7-10 days.
You can't dry chamomile in bundles hanging down; the yellow part of the flowers is the strongest part of chamomile, and the yellow bits fall off as the flowers dry.
You can't dry chamomile in a dehydrator, either, as those flowers are small enough to fall through the mesh of the dehydrator trays.
The usual test of dry herb, "stems break, they don't bend", doesn't work with chamomile, as the stems are brittle and break when fresh, too. Wait for the leaf to crumble. When that happens, use scissors and cut your herb into 2-3 cm bits (about 1") and put the lot into tight glass jars, to be kept in a dark cupboard until use. And remember to label the jar: "Chamomile, July 2005, Helsinki". Not because you would forget that this is chamomile (nothing else looks like dried chamomile), but because you would forget the date and place if you didn't label things. BTDTGTT.
Your dried chamomile should be green, bright yellow and shiny white, not the greyish yellow mess you get when you buy chamomile in the health food store.
Related entry: Chamomile