Genesis 9:3 Every moving thing that liveth shall be meat for you; even as the green herb have I given you all things. (KJV)
Romans 14:14 I know, and am persuaded by the Lord Jesus, that there is nothing unclean of itself: but to him that esteemeth any thing to be unclean, to him it is unclean. (KJV)
I Corinthians 10:25 Whatsoever is sold in the shambles, that eat, asking no question for conscience sake: (KJV)
Deuteronomy 14:7-8 Nevertheless these ye shall not eat of them that chew the cud, or of them that divide the cloven hoof; as the camel, and the hare, and the coney: for they chew the cud, but divide not the hoof; therefore they are unclean unto you. And the swine, because it divideth the hoof, yet cheweth not the cud, it is unclean unto you: ye shall not eat of their flesh, nor touch their dead carcase. (KJV)
These passages are best understood by knowing the old and new covenants. The Genesis passage referred to the way things were before the downfall of man. At that point, no foods were 'unclean'. With the old covenant (Deuteronomy), some foods were unclean. With the new covenant (Romans and Corinthians), foods were only unclean only to those who feel they are. These are different laws for different people, not a contradiction.