Hello and welcome to GardenWeb!

The Bleeding Heart die back problem is a fairly common one. I'm assuming that you have the pink kind of Bleeding Heart, which is called Dicentra spectabilis.

I have the same kind of Bleeding Heart planted in our garden in full sun for some strange reason. It was planted by the previous owner of our house, and I've been meaning to transplant it to a more shady area. Our Bleeding Heart dies back around July. We have a groundcover called Barren Strawberry (Waldsteinia fragarioides) planted in the same bed which covers over the Bleeding Heart nicely once it dies back.

Our garden has a very informal, "country" type of look, so the strawberry ground cover looks appropriate. But I know a lot of people interplant their Bleeding Hearts with ferns, hostas, and astilbes if their garden has a more formal look.
These aren't necessarily ground covers of course, but they get larger later in the spring and cover the Bleeding Heart dieback very well.

Also, we cut back the foliage of the Bleeding Heart once it turns yellow.

Hope this helps...