Most versions of the American accent are a bit nasal compared to other English speakers, so I guess that makes it higher pitched on average -- even if I wouldn't call it "shrill".
Also, there is no a neat separation between nasal Americans and other English speakers. Many Australians are also quite nasal, whereas black Americans (unless you count Steve Urkel) don't strike me as nasal at all.
On the third hand: I find the Irish accent is higher pitched than most, without sounding either nasal or shrill. But Americans often sound a lot like the Irish, so that might also make them higher pitched.
Also, there is no a neat separation between nasal Americans and other English speakers. Many Australians are also quite nasal, whereas black Americans (unless you count Steve Urkel) don't strike me as nasal at all.
On the third hand: I find the Irish accent is higher pitched than most, without sounding either nasal or shrill. But Americans often sound a lot like the Irish, so that might also make them higher pitched.