By "drain the swamp", I don't mean the extermination of the Islamic community, but rather to examine the incentive structure and see where it can be changed.
One big issue is the Islamic madrassas, and how they and the radical clerics are producing a steady supply of Jihaadists. The Saudis are behind a lot of the financing of these radicals. What do the Saudis expect to gain? The answer to that question could provide an answer to the problem.
One possibility is that the Saudis want to use their financed network to exert political influence beyond their borders. The threat of "civil unrest" in European countries with large Muslim populations (eg France, Germany) may be a significant factor in these countries decisions
If you follow the money that finances the radicals, and deal with the people behind the money, you may go a long way to calming things down. I'm thinking that Bush's long-term strategy is to secure iraq before dealing with the Saudis