If they don't play the politically-correct "Islam is peace" game, they get flack from the American Muslim groups, IMO. The backlash has been mentioned by more than one writer, and I imagine it's much worse on TV, as evidenced by Lou Dobbs' being blasted from all sides about his "miltant Islamics" comment on CNN. I think, in general, we underestimate the influence of these groups, not only with the press, but with politicians -- including those Congress and the White House.
We all know that Lou Dobbs was right on target, and if Bush had had Lou Dobbs as an advisor he would have less of a pandora's box on his hands. But Bush won't say this is a war on ISLAMIC terrorists, because "our friends the Saudis" and the PC knee-jerkers would've gone apes**t. You're exactly right, groups like CAIR and AMA (Amer Muslim Assoc?) have much sway, and get much Wahhabi money with which to hijack the mouthpiece...and all the while, the U.S. media and Katie Courics hold them in such serious esteem with slack-drawed awe and revery. CAIR has even gotten Daniel Pipes banned from the supposedly "fair and balanced" and tax payer funded NPR, all because they don't agree with the spotlight he shines on the islamofascists. Because Lou Dobbs has dared to do the same, he too is being targeted.