Matthew points to an interesting poll which suggests people in Islamic countries are more in favor of democracy than we in America and the UK. I think that this speaks more about the universal desire of human beings for freedom they don't have rather than any particular insight into Islam. Remember that in places like Bangladesh, "democracy" is synonmous with America in a semantic sense, and America is synonymous with the images of wealth and luxury and power that even the poorest slum residents see on Star TV.
But there is indeed a direct link between Islam and democracy, a positive one whose authority comes straight from the Qur'an itself. I've previously discussed this in the context of Iran and the struggle for freedom from the theocracts there - the essence is understanding that religious freedom is essential to Islam. Freedom of religion is enshrined in Ayat 2:256, which states that "there is no compulsion in religion."
In fact the Islamic argument against imposition of religious belief stems from the same philosophical root as the idea that proving the existence of God is counter to religion's self interest - both deny deny faith. If I oonly pray or wear a beard because of fear of the religious police, then what value is my religious action? Absolutely zero. You cannot be compelled to faith, it must call you to it of your own free will. And free will is the highest faculty of Man, the sole gift of God.