After the blooms are spent, the foliage looks OK for a while, but then my leaves start browning and look pretty ugly. Is there a way to prevent this? Should the foliage be cut back?
Generally, in CT peony foliage stays pretty much in good shape till October and only then start decline.
The only two reasons I could think of why yours foliage is not good, either you forgot to water them from time to time and they are going into early dormancy or they have a Botrytis.
There are plenty of info on a net about this fungus, but i include the link to the article that explain this particular and many other peony problems in a simple terms.
Here is a link that might be useful: Peony problems
It's probably Cladosporium. Heres a link that helps explain it..
Here is a link that might be useful: Cladosporium