In addition to Affable's answer, there is also Genesis 9:3 which indicates that at least mankind did not eat meat before the flood:
Every moving thing that lives shall be food for you. And as I gave you the green plants, I give you everything.
It is arguable that animals did not prior to the release from the Ark, as well - they also were not afraid of men until after (the previous verse):
The fear of you and the dread of you shall be upon every beast of the earth and upon every bird of the heavens, upon everything that creeps on the ground and all the fish of the sea. Into your hand they are delivered.
However, regardless of whether animals were carnivorous prior to the flood or not, we know that God provided for them via Noah's efforts to lay-up stores in the 100 years before the flood.
Noah did this; he did all that God commanded him.