The Qur'an says that whoever picks a religion other than Islam is of the losers and will be an inhabitant of the Fire. The four schools of Islamic law (I only know about Sunni's so I'm not too sure on the Shi'a) regard apostasy as grounds for execution.
However, there are alternative views. My mother argues pretty strongly that:
-There is no temporal punishment mentioned in the Qur'an for apostasy, in the case of other crimes such as murder suitable punishment is given (varying degrees of as a matter of fact, not all cases of theft must end with a dudes hand being cut off).
-The word "Islam" means "submission to non other than Allah (God)", it does not neccesarily refer to the religion founded by Muhammad. Just like Abraham is not a Jew or Christian, (how could he have been?), he was a Muslim, the word meaning "That which submits to the will of God". As a result Christians, Jews, Bah'a'i could all be described as Muslims. The same is also true with Bhuddists, Hindus and Shintoists, pretty much all religions submit to some concept of God.
-In the Hadith there are accounts of Muhammad executing apostates, however, in these that was not the main cause. The main reason for their execution was that they had gone over to the pagans sides. In the early days, Islam was a religion, a state, an army, a way of life. They were active enemies of the Ummah and so the punishment was more political than anything.
Of course, this is seen as a lunatic, fringe view within the conservative elements of Islam.