I usually chop my dead coneflowers after it's obvious they're done for the year. I've heard that others leave them up for the "winter interest" or for the birds, but I've never thought they were particularly attractive once dead, nor do I want to help birds.
See, I'm terrified of birds. I won't go into how many times I've been bitten by those bastards, but yeah - anxiety level goes through the roof when I'm around them or see them. So to feed them and keep them around my yard? Hell no!
However, something obviously got into me this year because I left my dead coneflowers out. It was an experiment of sorts, to see if they'd actually take the seeds and if I could live with them doing so.
Well, they obviously filled their evil little beaks and I didn't once see them do it. And that is totally fine with me - I can coexist that way. I'll get some good karma points by leaving food out as long as they don't let their presence be known.