I would nominate Radiohead on a technicality. They're generally thought of as a 90s band, but 2 of their biggest records came out in the 2000s (Kid A in 2000 and In Rainbows in 2007).

From the more obvious picks, I'd nominate The Strokes, Paramore, Florence + The Machine, Portugal. The Man, Coheed and Cambria and The War on Drugs