Nope.
As you'll see by my signature, I don't even have a Nord character. I play Dunmer and Khajiit only, and I don't feel weird about it at all. The theme/story of the game works regardless of race in my opinion, and I like the way my Dunmer and my Khajiiti characters fit into it.
On a side note, across Daggerfall, Morrowind, and Oblivion I
did give the other character races a chance. But none impressed me or sparked my attraction for a character like the Dunmer and the Khajiit do. Which is why ever since Daggerfall (my first TES game) I've had my running Dunmer character, and ever since Morrowind I've added the Khajiit into the mix.
But now, I don't even try other characters. I've given them a chance, and they failed to impress me. So it's Dunmer and Khajiit only for me.
And as I said before, I think that the story, themes, and plot lines work well regardless of character. Maybe it's the thinkly veiled racism against everything not-Nord in the game that makes you think that, but I happen to know that my characters fit in just fine in Skyrim.
