I'm no expert, but I have the same issue, living on extremely heavy soil with a high water table. The yard slopes slightly, and there's an old overgrown "ditch" running down the lowest point, 20' before the gravel platform of the house (which drains nicely).
I've experimented with two different ideas over the past 2 years and I'll outline the experiences below.
1. I cut a 9-12" ditch (about 6-8" wide) running down one side of the hill, slightly on contour, but mostly along the edge of the "wild island" (which is an overgrown patch with old-growth stumps and such, which we kept for the wildlife habitat) as it was more convenient. This helped with drainage along the one side. Ideally, I would have cut a ditch on contour, feeding into the first ditch which runs down the hill. I didn't want to cut the ditch on contour, because this would cut the usable/cleared part of the side yard in half. I support in retrospect I should have added a french drain style ditch so as to not interrupt the usage too much. The main problem with this would be the grass. Grass roots would likely fill the french drain within a year and I'd have to be actively maintaining this ditch. I guess it might be worth it though...
TL;DR: Dig some shallow ditches as on-contour as possible.
2. When removing patches of sedge, and buried trash that the previous tenants buried/left in the yard, I was left with some holes. Some deeper than others. I patched these holes using pebbles and gravel from the beach across the street. A few of these holes happened to be near a low/wet spot in the yard, and once they were filled with gravel, the areas are no longer soggy after the rain!
TL;DR: Dig a pit in the wet spots and fill with gravel/sand/pebbles to enhance the drainage into the soil below.
As I'm writing this out, I'm realizing a combination of the two would suffice. I'm inspired to go work on this now...