From: aks3.cornell.edu (Amy Smith):
You don't need anything terribly high tech to dry herbs...
For leaves you can use old window screens in a DRY DARK place (like the attic). If you are growing seedlings, place the screens on the top of the fluorescent lights.
For roots you might want to use the oven on low. Chopping the roots first helps. Food hydrators are better for roots.
Flowers are like leaves but you MUST be careful about keeping it dark. (the sun leeches the nutrients out of herbs and they disintegrate too.)
You can also hang bunches of leaves on string in a dark place outside (if you are in a fairly low humid climate) Or put flowers or leaves in paper sacks and hang them to dry (shake them or stir them periodically so they don't stick together inside the sack.)
Never store herbs in completely airtight containers unless you have access to a desiccant (like you get in pill bottles) to store with them since you will never perfectly dry them and therefore they need to breathe.
From: jrogow.ridgecrest.ca.us (Judith Rogow), in response to above:
An old remedy - tie a few grains of rice in a bit of cheese cloth and add it to the bottle if you must keep it tightly closed.
Never use a microwave to dry herbs. First, timing is different for different microwave ovens, second, the taste isn't that good, and third, you can end up with a fire in your kitchen.
If you use a dehydrator never go above 40 deg. Celcius. Most herbs are tasty because of volatile oils and in high temperatures these volatile oils get volatile and your herbs get tasteless.
And, if you use the bunched-herbs -method outlined above, strip the leaves off the stalks after your herbs are dry. Stalks aren't that tasty in soups, and can be tossed on barbecue coals to give some taste there.