Peonies and brown spots

[seamlyne]

Now that the peonies are done blooming, we'll have the greenery for the rest of the summer. Seems like every year, though, the leaves start to get brown and smell...well...bad. My wife insists that's Just What Happens, but I'm not convinced. I know they eventually dry out, but this seems more like a rot or fungus.

Is there a way to keep the peony plants green for the rest of the season, or is this just how it works?