Muslims may not appreciate (or perhaps they would) this distinction. But Hilare Belloc claims in The Great Heresies, that Islam is a heresy of Christianity.
Therefore, if Mohammed had had a vague inkling of Christianity and claim the inheritance of Ishmael then yeah, of course they worship the same God.
However, we couldn't say they worship Him fully because they do not believe in the Trinity. They do not believe that Jesus could be God.
In response to Caleb's answer wherein he says that no Christian sect, except liberal traditions embrace Muslims as believing in the same God.
From the documents of Vatican II (not just something a talking head says):
The Church regards with esteem also the Moslems. They adore the one God, living and subsisting in Himself; merciful and all- powerful, the Creator of heaven and earth, who has spoken to men; they take pains to submit wholeheartedly to even His inscrutable decrees, just as Abraham, with whom the faith of Islam takes pleasure in linking itself, submitted to God. Though they do not acknowledge Jesus as God, they revere Him as a prophet. They also honor Mary, His virgin Mother; at times they even call on her with devotion. In addition, they await the day of judgment when God will render their deserts to all those who have been raised up from the dead. Finally, they value the moral life and worship God especially through prayer, almsgiving and fasting.
Nostra Aetae - From the Documents of Vatican II
With reference to Muslims, it states these few things which are common to our faith and conveniently takes a distant reference from Abraham which is a common point on worshiping same God. It makes no reference to Mohammad though he is the epitome of this religion. Again, no recognition of Muhammad, the Qur'an, and the correctness of what Muhammad taught. So, as becomes clear in other Church documents, whatever truth there may be in what Muslims believe must come either from the universal truths that God has always made accessible to the minds and hearts of all people, or else by transmission from the divine revelation granted to the Jews and Christians.
So, yeah, even though this questions wasn't directed at Catholics in particular, it is true that most Christians (and the entire Communion of Saints) believe that Muslims attempt to worship same God, but they do so in severe error.
and, just so no one thinks this is out of context, the document says of Islam and other non-Christian religions
The Catholic Church rejects nothing that is true and holy in these religions. She regards with sincere reverence those ways of conduct and of life, those precepts and teachings which, though differing in many aspects from the ones she holds and sets forth, nonetheless often reflect a ray of that Truth which enlightens all men. Indeed, she proclaims, and ever must proclaim Christ "the way, the truth, and the life" (John 14:6), in whom men may find the fullness of religious life, in whom God has reconciled all things to Himself.
My pastor explained it pretty succinctly to a Religious Ed. class who was doing some Q&A with him. He said it's like a theatre where everyone's looking at the same play but quite a few people (non-Catholics and non-Christians) have for whatever reason decided to watch the play from awful seats instead of getting the front row seats that they could have had, had they only desired them. That's why I said above, they do not worship Him fully, but they do worship Him.