My guess is that blueberries, blackberries, and gooseberries (roughly in that order) are the fruits on your list with which you'll most likely have success.

Your summers may be hotter than currants and gooseberries would prefer, but they may still work for you. The spot you described that only gets morning sun might be the best compromise for a Nashville summer.

Blackberries can be a lot tougher than raspberries, and raspberries would probably prefer cooler summers, too. If you want excellent tasting blackberries that are easier to grow consider some of the thorny varieties. The thorns are a very significant compromise, but the thornless varieties seem to compromise taste, vigor, and or ease of management (particularly as far as training/support).

With blueberries be sure to consider which type (rabbiteye, southern highbush, or northern highbush) does best in your area. I suspect it will be rabbiteye. Soil acidity (pH) is also especially important to consider with blueberries. If soils in your area are generally acidic, you likely won't have to do anything special, especially not for the apparently slightly more adaptable (with regards to pH) rabbiteye types. For a long season the rabbitye varieties I'd consider would include Yadkin, Premier, Bonita, and Onslow.

You might also consider serviceberries/juneberries. Disease (specifically rust, possibly associated with red cedar trees in the area -- this is my best understanding anyways) can take a heavy toll on serviceberries (some species perhaps more than others) in some locations, but it's not a problem at all in other locations, and in those locations serviceberries can be super easy to grow.

There are also some smaller natured fig varieties that could be pretty easily be kept to a modest bush size but figs will die back in the winter if it gets much below 10 degrees.

You could get blackberries from June to October with the following:

(Early) Osage or Ouachita
(Mid) Triple Crown
(Late) Prime Ark 45 or Freedom

Blackberries will fruit in partial shade. As will alpine strawberries and if your soil is acidic enough, evergreen huckleberry.

SHADE TOLERANT (Raintree nursery list)

Partial Shade
May have less fruit.
Alpine Strawberry
Bamboo (some types)
Dogwood
Elderberry
Evergreen & Red Huckleberry
Filbert
Gooseberry
Hydrangea
Jasmine
Lipstick Strawberry
Lingonberry
Magnolia Vines
Musk Strawberry
Paw Paw
Pie Cherry
Salmonberry
Serviceberry
Sugar Maple
Thimbleberry
blackberry

Full Shade

Bamboo (in sunny climates)
Currants
Carolina Allspice
Evergreen Huckleberry
Flowering Currant
Wasabi
Highbush Cranberry
blackberry

This post was edited by Fascist_Nation on Tue, Dec 30, 14 at 21:13

I'm about 70 mi NW of you.
Plant rabbiteye and Southern highbush blueberries - don't bother with the northern highbush, and certainly no lowbush types unless you just don't want to be bothered with a lot of berries.
Kiowa blackberries wore me out this year - picking every other day for what seemed like two months... heavy production of HUGE berries with good flavor - if you let them get fully ripe. But, they are thorny as all get-out; worth a little blood sacrifice, though. Not a raspberry fan (blacks are ok), so I can't give you any guidance...though Baba berry is well-reputed to do well in warm climates.
Getting warm in these parts for gooseberries. Pixwell and Glenndale will work - even in full sun, but if you have a spot with at least afternoon shade, you might get some better quality varieties to do OK.
Currants...I don't know...do you even know if you like them? (I keep hearing 'cat pee flavor/smell' on the blacks). If so, a semi-shade or at least afternoon shade spot might work.

Susan,
Nashville I believe is USDA Zone 6a so when shopping for plants you can look at the plant hardiness zone. As an example zones 4-9 are the zones Anne raspberry can survive and since 6a falls within that criteria it would be one to consider. In the description it may or may not say full sun or partial shade etc. as well. Gooseberries and currants like some shade so you might consider growing those in your shady spot.

Here is a link that might be useful: Anne Rasberry for example of Hardiness Zones

Nashville TN is in UDSA Hardiness Zone 7a.

Thanks jtburton it appears they changed the hardiness there a couple of years ago to 7
http://plannedresilience.net/new-usda-plant-hardiness-zone-map-for-2012/

This post was edited by ClarkinKS on Sat, Jan 3, 15 at 6:06

Thanks so much for all this information!

You could try constructing a small shade screen to protect berry plants from late afternoon sun. They have screens that let in 30% sun, or 50%, or 75% etc...
Probably wouldn't be too difficult.

You might look into muscadines.