I have a different opinion. Fungi, mold and microbes are everywhere. While it might be good to move the peony foliage away from the peonies, carting it off or burning it might be over-kill. Soil scientist and composting scientists are saying that we need a balance of microbes and stuff, not a complete absence of bad stuff. A bunch of greenish peony foliage might heat up a compost pile, especially if you could add some browns to it too.
Personally, I would do whatever is easiest, but piling foliage up with cardboard layers and just letting it sit for over a year should be just fine from what I read.
One soil scientist said that you get in trouble if you only have browns or greens and let it sit, if you have a mix, it works. I'm guessing even pine straw or bark with the peony foliage would work too.
Our previous owners just piled stuff up at the wood line and was it lovely dirt there!
