I remember when I was a YEC (young earth creationist), whenever presented with the similarity of DNA between species, the common counter-argument to a common ancestor was that similar organisms would have a similar DNA structure because they were similar in shape. God was basically reusing a good design. So humans and chimps would have similar DNA because they had a similar body shape (because God created them that way) and not because they shared a recent common ancestor. At a high level it makes sense, but I have a hard time answering several questions.
Why would a cow have more genetic similarity to a whale than a horse? Or what about the Hippo? The hippo has more genetic similarity to a whale than it does to pigs or cows. Clearly, the similarity in DNA does not dictate a similarity in form. I am trying to find an article that is easier to read, but here is the original scientific paper. Through fossil evidence scientists had figured out that whales had descended from hooves animals, specifically ungulates. Cows are ungulates, but horses are not. A good resource to learn more is Carl Zimmer's book, At the Water's Edge: Fish with Fingers, Whales with Legs, and How Life Came Ashore but Then Went Back To The Sea.
Why do we have a disabled gene in our DNA that is used by birds and reptiles to make egg yolks? Not only do we have that gene, but when we look at the DNA for a chicken, that same gene is in a fairly similar region. Again the paper is hard to read if you aren't used to reading scientific papers.
Why do we share fragments of viral dna with other apes? Certain types of viruses use their host's DNA to replicate. Occasionally the DNA strands get mixed up and the virus's dna is incorporated as our own. There was a recent article that was discussed by two bloggers, Ed Yoong and ERV. Basically new research is out estimating that 8% of our DNA are viral leftovers. Now the freaky part is that there are numerous sequences that we share with apes and not only do we have the same viral leftovers, but they are in the exact same positions in our DNA. I first learned about this on Talk Origins.
Now I see all of these as strong evidence for common descent. If at any point evolution would be proven wrong it would have to be with the discovery of genetics. But it seems to me that the case for evolution is only stronger.