I have the same problem with the acid-coffee tummy.
Acidity in coffee most often comes from the soil environment in which it's grown. Brewing processes can increase or decrease the amount of acid that is extracted in the process, but if you start with a high acid coffee, you'll end up with a significant acid level even if you cold brew (as Cheflayne suggests.)
Brazil and Colombia produce the lion's share of coffee that reaches the mass market (Folgers, Maxwell, Costco etc.) and both of these are very high acid coffees. Most large commercial coffee roasters base their blends on Columbian beans mixing in other varietals for flavor.
The lowest acid coffee I've found is Sumatran and some other spots in Indonesia. Others grown in highly active volcanic areas seem to have lower acid contents too-Jamaican and Hawaiian to name a couple others.
We sell a lot of Sumatran French Roast in our shop - big punchy flavor that's very smooth and easy on the stomach. To me, it's a little flat in flavor and I like to mix it up with some other coffees with a touch of acid to brighten up the flavor.
Dean's Bean's-out of Boston has a delightful blend of dark and light roasted coffee called Mokka Sumatra you should try. We sell tons of it and I don't know anyone who doesn't love it. Also Larry's Beans (another small, organic and fair trade roaster) sells one called Cowboy blend that is terrific and also easy on the stomach.
RE: Brewing process- avoid French press in this case. The longer the coffee sits in hot water, the more caffeine, acids and other VOCs are extracted from them. Cold brew is great because the cold water process does not extract much of the acid and you get a really smooth cup-great for iced coffee too.
Cold brewing results in a thick concentrated coffee syrup that you add hot water to for a hot cup or pour over ice with milk for the best iced coffee ever.
Counterintuitively, espresso drinks tend to have big punchy flavor without a lot of caffeine or acid since steam is forced very quickly through the coffee grounds extracting the flavor compounds but not a lot of the other stuff that can be problematic to some people.
On a personal note- organic, shade grown, and fairly traded coffee only costs a little more than the grocery store stuff. If you buy your beans from a knowledgable independent coffee shop owner who focuses on fair trade and organics, your $$$ help small business people and their families all down the line-from the farm to your cup.
Go to Dean's Beans website (http://www.deansbeans.com/) to read about Dean's efforts in the developing world to help coffee growers and their communities. His Java Trekker stories are really interesting. Nice guy too.