They're not priests and it's not a misinterpretation of Islam. Read the books, folks. Mr. Blair, like Mr. Bush, is still playing realpolitik. It's an understandable, but increasingly untenable, position.
The fact is that violent, offensive jihad was declared on the kuffar fourteen centuries ago, and a robust jihad revival got underway when the last remnants of the Uthmani (Ottoman) Khilafah (Caliphate/Islamic state) were dismantled by the West in 1924—a year that Muslims worldwide remember vividly, but one that most of us don't think about very much.
Also, Speak Easy links to Mark Steyn, who notes that "Islam has become the world's pre-eminent incubator of terrorism at its most depraved".