The most potent political force in the Middle East is not Islam but Arab Nationalism, and that's what the insurgents are fighting for. Religion gets thrown in the mix because of a perception that "Muslims" are hurting because of the Western-Israeli unholy alliance.There are of course some Muslim radicals who have the formula of "If you infidels-Crusaders-Zionists do this to us, then by applying certain verses of the Quran, we would do WHATEVER is necessary to stop you," but that reactionary ideology has been triggered by fear that Arabs and Muslims are under attack, not Islam per se.
If you go and live in the Middle East, they always use the term "Arabs and Muslims" to identify themselves. The word "Arab" always comes first which has led some non-Arabs to ask, "Are Arabs not Muslims?" First and foremost, Arabs are seeking their political independence from the West, not the establishment of the Caliphate, but simply freedom from foreign domination. Yet, they do not know how to achieve that goal. They don't know how to organize themselves politically because they've never known democracy. When Americans claim that they want to help and be the instrument to bring freedom and democracy to the Middle East, Arabs see that as another cunning Western trick in order for them to keep their hegemony over the Middle East.
They Iraqi insurgents don't care if there is democracy or not, as long as Arab Nationalists end up controlling their own land and resources-not Americans, not weak Muslim rulers under the patronage or protection of Western Powers. The ordinary Sunnis, on the other hand, simply want the US to leave. They want democracy, as the largest Sunni party has always claimed, but without the help of the US.