It took me a good bit of sleuthing, but I've finally determined these plans are probably T. cuneatum (Toadshade trillium) There are two of them growing in full sun near the edge of a lined pond, each in the middle of a clump of liriope.
I understand these plants do best in full to partial shade and would like to move them, but when and how? Do they die back in the summer (I don't remember seeing them last year, but I could easily have overlooked them back then.)
These plants are not likely to win any beauty pageants - at least not on their "natural good looks", but the more I read about them, the more intrigued I am. I'd like to make sure they make the move okay. Any help is appreciated!