The major reason you put your tea bag in hot water is not to cook the tea, but rather to speed the process of dissolving useful compounds out of the tea leaves or herbs into the water. Water doesn't have to be hot for this to happen, but the cooler the water is the longer it takes to dissolve a meaningful amount of the compounds you want.
What I've started doing is harvesting my own tea herbs, and drying them at room temperature and out of the sun (sunlight makes some compounds in some herbs more available, but breaks down most compounds in most herbs, rendering them less available and/or less medicinal). Once the leaves are dried, I store them in airtight jars in a cool, dry, dark place.
When I want to consume them, I'll fill a jar with water, put in the herbs, and leave it out for a day or so, until the water takes on the color of the tea. If I want something warming, I'll pour this (straining out the bits of herbs) into a pot and warm it slowly until it's above my body temperature but not hot enough that I feel like I'm at risk of being burned while my finger is in the warming water. I've found that this is warm enough to give me the warming feeling I used to associate with hot tea, while still keeping the temperature low enough to avoid cooking the herbs.
Finally, although I have a lot of herbology books, I'm becoming skeptical of a lot of the claims made regarding the medicinal properties of herbs. I think placebo effect plays an enormous role much (though not all) of the time. I've also met several herbalists who doctor themselves with all their fantastic concoctions, and many of them just aren't healthy people. That alone makes me wonder...
Still, some herbs are very nutrient dense, and are worth keeping around for winter when fresh green plants are gone. I stock up on dandelion root, burdock root, red clover flowers, and nettle leaves. The roots and the nettle leaves are among the most nutrient dense plant foods available in my part of the world, by virtue of their life strategies (i.e. send deep taproots into the subsoil to draw minerals up that most other plants don't gain access to). I have to work for the dandelion and burdock roots, but nettle grows so abundantly that i can fill a pillowcase with fresh leaves in an hour. Clover flowers yield an antioxidant rich, sweet tea that just tastes nice, and the flowers are very easy to gather, like nettle leaves.