There are, of course, violent radicals in every branch of religion and atheism. But, Islam has more than its fair share of such people.
War is not entered into lightly; when examined on a global basis one needs to take a suitable time frame and not one centered on the present moment/decade. For example, one might choose the whole of the 20C; Hobsbawms, a famous Historian, chose this time period in his book Age of Extremes; this showed the enormous turbulence in Europe caused by the radicalisms of Nazism, Fascism, Marxism & Anarchism in a globally dominant West in the wake of the collapse of the Ottoman Empire; then the loss of colonial possessions and the resulting pax Americana.
More recently there has been two great wars in Africa involving nine nations centred on the Congo due political instability post (European) decolonisation.
One ought to conclude from this that violence (as well as peace) is the lot of nations and not that "Islam has more than its fair share of such people".
Its also worth noting that the cold war division between Western Europe & Eastern Europe was marked by the the Western Capitalist 'Free World' versus The Eastern Communist 'Evil Empire'; whats less noted is that the division also falls between Catholics/Protestants and Eastern Orthodox in terms of the Christian religion. Is this important - possibly; or it may show that the argument is not religous, but ideological; or even not ideological but geo-political.
Islam is not against Christianity
Agreed. It simply has theological differences with it; as there are indeed within Islam itself.
For instance, nowhere in the Quran it teaches the Christian Trinity is false
Islamic theology is explicit that Allah is One and indivisible (tawhid); Catholic theology is centred on the Trinitarianism - where God is three persons (coexisting consubstantially in One being); but notably there are minor positions like Unitarianism which adopts a position similar to Islamic tawhid. Darwin, Newton & Emerson for example were Unitarians.
My question is, then, why are so many muslims taught that violence against other religions is the way of God?
The argument is against other religions per se; the argument should be centred on the geo-politics; but one should note that political identities are intertwined with religous ones; for example Iran, a regional power is mainly Shi'ite.
The Middle East has been central to geopolitics globally for securing energy policies; and has been a theatre of European interests until the collapse of European stability after the second world war, and whose positions were then taken up by America as the globally dominant & western power. This explains in a very crude way the regions relations with the outside world.
One then needs to take into account various regional & religous rivalries - for example between Sunnis & Sh'ites; and many other factors that I'm too ignorant of to list here.
Why do muslims not speak out against these teachers and show they are wrong?
The Pakistani Islamic Scholar Tahir ul-Qadri has written a Fatwa against Terrorism to underline that terrorism is illegal under Islamic law, and grounded his argument in the Sunnah & the Qu'ran. It has been endorsed by al-Azhar university:
"The special committee of Islamic Research Council, Al-Azhar commenced its [the Fatwa’s] examination and scholarly review and found that the author’s book discusses terrorists who reside in Muslim states...are like the Kharijites who appeared during the time of the Messenger. The author quotes the opinions of Islamic scholars who considered it a religious duty to fight and kill the Kharijites if they refused to renounce their doctrine after receiving warnings and advice.
The author identifies the terrorists in Pakistan as rebels and purveyors of corruption, and contends that their traits and actions correspond to the Kharijites.
The author maintains that these traits put the terrorists at odds with the main body of Muslims; and he considers it obligatory to warn them and advise them and, if unheeded, to fight against them until their threat subsides.
This examination committee maintains that the author’s conclusions are in agreement with the principles and directives of Islam, especially considering the fact that he furnishes his judgments against them with proofs from the Book [Qur'an] and Sunna detailing their traits and actions".