Alright. Here is a more comprehensive answer.
In the times of the Old Testament, people were required by God to give animals to the High Priests to be sacrificed and eaten. However, after the Crucifixion of Jesus this practice was no longer necessary.
I think vegetarianism and veganism, if followed as a discipline, can be pleasing to God. God likes it when we give up things in his name and there is a tradition of doing this in the Church (the vows of priests and monks, and practice of giving something up for lent). If your motive for being vegetarian is to honor God by having a simpler diet then I think it is alright.
It is a problem, however, if it leads to attributing an undo status to animals. While God created animals and said that they were good, he did not make them equal to humans. We alone are known to have souls that persist beyond death, and when God came to Earth he did so as a human.