In the Charlotte, NC area, Harris Teeter, Winn-Dixie, and The Fresh Market accept Amex. Bi-Lo (our local Ahold store) and Food Lion don't. We're also now getting Lowe's Foods (no relation to the hardware store), but I don't know their Amex status.
During my trips to the Long Island/NYC area, I've found that Waldbaum's (an A&P-owned store) is an 'Amex friendly' market.