As mirfak said already, I think the pronunciation can go either way in the U.S.
I knew one
Aimee who pronounced it AY-mee; she had a French last name but did not speak French or, as far as I knew, have any French near relatives.
I knew another girl who had
Aimee (with the accent) as a middle name; her first name and other middle names (she had more than one) were all French, and her mother was a French teacher who had grown up in
France. Her "
Aimee" was pronounced ay-MAY.
IrishGirl mentioned that the presence of an accent indicates the French pronunciation, while its absence indicates the American. I think this is often true, but I wouldn't use it as a rule.