Maybe Liriope, but in full sun it will probably need some supplemental water in drought to look it's best.

You might want to check out groundcover for sunny dry area and more suggestions.

Not sure I understand. Do you have in mind to put a ground cover in your beds, with your shrubs and perennials, to suppress weeds?

Thaks for the suggestions, LT.

Whitecap...let me explain better.

I'm at the end of the street, at the middle of a cul-de-sac, with a pie-shaped lot. The front yard isn't very big, maybe a bit larger than some zero-lot-line houses, but the backyard is HUGE.

The neighborhood was built in the early 60's. The section I'm talking about has the driveway to the north, house to the west, sidewalk to the front door on the south, and city-sidewalk to the east. I'm removing all of the grass from that section and part of the section on the other side of the walk to the front door, south of that walk. I want the shrubs and perennials with something planted as a groundcover/living mulch/whatever in between. There will be "swaths" of groundcover, partly to keep from having to keep adding mulch and partly because there's too much ground to NOT have anything more than mulch there.

I don't care if it's a native that "everyone" calls a weed, like horseherb or something along that line. I just need something to help keep the soil where it is (there's a slight incline from the house to the street, enough for water to run down the concrete) and to "look nice". The front yard doesn't turn into a lake like the backyard, the house blocks the rainwater runoff.

I don't want mondo grass or asian jasmine, both are way too over-used and I don't want to deal with either. Sorry. I figure I'm already being different by removing the grass, might as well go all the way!

Blue shade ruellia - mine's in mostly shade but looks very nice. Dies out in winter but returns.
Frog fruit- it can be mowed periodically to keep thick
White rain lilies - the evergreen ones
Ice plant - I have Kelaidis. Thick, ground hugging, evergreen, pinkish apricot flowers. Probably not great for large area, but could be used in tadem with other groundcovers.

If you find something other than Asian jasmine, let me know, and don't even think about using that stuff as a "living mulch" unless it's contained apart from your perennials and shrubs. The reason you see so much of it isn't because people are unimaginative, but because of the absence of other hardy, low, full sun groundcovers. Blue shade does need shade, as does mondo.

Perhaps Wedelia. Daddy "helped" himself to a piece of some growing at a Toyota shop.
He always called it Toyota plant.

You might be a tad too North for wedelia. But I grew it here in San Antonio.

Here is a link that might be useful: Wedelia

Someone asked me the other day about my frogfruit. Thought I'd post an update. I planted a few sprigs in a 'hot spot' garden next to the driveway and sidewalk. I wanted it there b/c I knew it could get trampled and bounce back, and could take the heat like a champ.

Here is how this spot looked last summer. You can see it still looks pretty weedy. I hadn't cleaned it up real good yet. Lots of bermuda and other stuff:

And here is how it looks now. The low stuff next to the concrete with the white spots is the frogfruit in flower.

Tex-Ag-95, this plant might work for you. It spreads by runners overground, which makes it easy to pull back and control. It would need to be edged periodically, as I think most groundcovers would. I like it b/c it stays evergreen and during the winter when the salvia, etc die back the ground cover stays green. And it really doesn't need a lot of water. Mine gets watered, but I've seen this plant growing on unattended roadsides, pure sand, etc. It's tough, but I don't find it quite as invasive as bermuda.

I also have the wedelia, which I love, but it is a little wimpy in heat and requires attention even in dry shade.