We probably need to know more about your site and about your needs to be able to recommend things.

Is the area you are talking about sunny, shady, or in between?

What kind of soil do you have?

Is the area well drained, frequently moist, etc?

How much time & effort are you willing to spend for maintenance?

Are you willing to spray regularly for pests and disease?

Do you have any specific types of fruit in mind?

Are you looking more for general types of fruit or specific cultivar recommendations?

Sure! Answers below...

Is the area you are talking about sunny, shady, or in between?
It gets about 6 hours of full sun from mid-morning to afternoon, and is then shaded after that.

What kind of soil do you have?
This area was wooded and covered with leaves for years, so the soil is pretty dark with not a lot of clay.

Is the area well drained, frequently moist, etc?
Well drained.

How much time & effort are you willing to spend for maintenance?
Not a whole lot. I want to plant something, pick its berries, and maybe prune it once a year without much fuss.

Are you willing to spray regularly for pests and disease?
Yes.

Do you have any specific types of fruit in mind?
Mostly berries.

Are you looking more for general types of fruit or specific cultivar recommendations?
I was looking for general suggestions, but if you know of a particular cultivar that has done well for you, feel free to share.

I'd definitely try some pawpaws (Asimina triloba)! I really enjoy good tasting/high-flesh-to-seed-ratio cultivars. Here is a link to look over that will give you some ideas about great pawpaw cultivars:
Nolin River Nut Tree Nursery
Hidden Springs Nursery is a more local possible source:
Hidden Springs Nursery

You might consider currents and gooseberries (Ribes spp). A couple of people in this forum have had good luck with them. They'd be best with a little evening shade (like you have) in this part of the country. Also, these may require watering in the hottest/driest parts of summer. I'll leave it to the current connoisseurs to recommend varieties.
Hidden Springs Nursery

Magnolia Vine (Schisandra chinensis) would be an interesting choice. Note that the species and most cultivars are deciduous, so you'll need a male and female plant or possible a self-fertile cultivar.

Wineberries (Rubus phoenicolasius) often get good reviews. One downside is their possible invasiveness. They're rated as a "lesser threat" along with plants like English Ivy, Burning Bush, and Rose of Sharon by the TN-EPPC. Covering the patch with bird netting and keeping their vegetative spread in check would completely eliminate this concern.

Native persimmons (Diospyros virginiana) would be another good choice. Here's another link that will give you some cultivars to consider:
Nolin River Nut Tree Nursery
Here's Hidden Spring's selection:
Hidden Springs Nursery

Cornelian cherry (Cornus mas) is a good option.
Hidden Springs Nursery

Triple Crown blacKberries is an easy one to take care of in this area. Most dwarf fruit trees will do fine here too. =)

Wow, Brandon, thanks so much for your recommendations. I had never heard of Cornelian Cherry until I saw it in Organic Gardening magazine a few weeks ago, and I fell in love with it! Glad to know it will do well around here.

Also, thanks tn_veggie_gardner. I didn't know that there were dwarf fruit trees. I obviously have lots to learn!

Many types of fruit trees are available in dwarf (read "slower growing") versions. Dwarf trees will require about the same amount of light as their larger counterparts. The type of fruit tree that first pops into my mind when I hear the word dwarf is the apple tree (although there are many others). Apples may do alright, but the scenario you describe would be borderline and might not provide enough light for them to do as well as if they received more light.

The three reasons I didn't recommend apples, pears, sweet cherries, sour cherries, etc were 1) the amount of light available is near the lower limit of what these trees need to be able to do well, 2) you said you were mainly interested in berries, and 3) these trees generally require more maintenance (spraying, pruning, etc) than the other choices I listed above.