I am thinking about similar thing. You could google for "knot garden." That could be another option. You can create simple of complicated patterns with the herbs that you mentioned and grow some flowers in some of the spaces too.

Here is a link that might be useful: Some examples of knot gardens

Wow, those knot gardens look really cool... a bit too formal for my yard, though.

I want mine to look a little more "wild".

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I was thinking the same thing too. I will "borrow" the idea and form some patterns in my garden with herbs, but probably not as formal as shown in the pictures.

I've been working on growing herbs from seed over a large area (3000 sq. ft.). My project is different from yours in a couple of ways: 1) I'm growing taller herbs than you're proposing, in the hope that they'll dominate the inevitable weeds; 2) The area in question won't get any significant foot traffic. So my project is more of a "meadow" than a "lawn". It's partly sunny and partly shaded. Nonetheless the lessons I've learned may be useful to you. Note that I'm in Maryland. You don't say where you live, and your conditions may vary greatly.

First try: Tilled and raked the area with a tractor, as one would do for a grass lawn. Broadcast the seeds in the spring, as soon as I thought the soil would be warm enough for the seeds to germinate. This was pretty much a disaster. For one thing, I didn't see any seedlings, I suspect because herb seedlings aren't robust enough to root in a firm lawn-like seedbed. For another thing, the necessary watering encouraged crabgrass, which quickly covered the ground. I also had problems getting some parts to stay moist, maybe because the water ran through too quickly.

Second try: I addressed the moisture issue by spreading and tilling in 16 yards of compost. After tilling, I didn't rake, rather left it loose, and walked on it as little as possible. This time I seeded in the fall. Got some pretty good germination, but there were a couple of problems: 1) Had to reseed one part, after I wasn't vigilant enough about not letting it dry out; 2) It was hard to water evenly, and some parts got too wet, which ruined the seeds there.

That was last fall. Now the area is carpeted with winter weeds, some of which I've pulled out (big area to pull weeds out of). But there are some signs of the herbs starting to poke out of the ground. It's been unseasonably cool so far this spring. I'm hoping that when it gets warm, the winter weeds will have had their day, and the herbs will take off. I don't think the herbs will do very well without water, so I expect to water them in the spring and summer. I also did plenty of watering last fall. I have a well and don't pay for water.

Herb seeds are small, and you have to mix them with a medium in order to spread them. I used sand as the medium for the smaller seeds, and cat litter for larger seeds. I bought seeds in bulk (which can mean 1 oz. for herb seeds), in amounts that would yield me roughly 10 seeds per square foot (i.e. 30000 seeds for the 3000 sq. ft), since I knew there'd be a lot of loss.

Anyway, it's a challenging project, but you could end up with something different from everyone else on the block.

BTW, if it doesn't work out this year, and I have to try again, I'm thinking of planting a "nurse grass" along with the seeds, in the fall, to keep the weeds under control while the herbs are getting established.

I actually have grown just clover in a lawn. I would suggest white clover, since it grows shorter and will never require mowing. Rabbit foot clover is nice too if you can find seed. For thyme, I would suggest a low crowing creeper called wooly thyme. Some might cringe at this, but you might also consider growing dandelion (good eating, good medicine, deep tap roots, pretty flowers).

lol - the thought of growing dandelion makes me smile. There's a large section in my yard that is never mowed, and throughout spring and summer, there's a beautiful progression of wildflowers. Dandelions are one of the many, and there are a couple of weeks when the entire area is bright yellow. But I'd love the area to have lower plants so that I could walk through it as well. I love the looks of creeping thyme but haven't tried growing it yet.

It would be interesting to have an update from torajima now, 3 years down the line. My guess is that mostly weeds filled in the "dirt."
It would have been helpful to know WHERE this yard is--growing zone, geographical area (don't need a specific address, just a general idea) so people can make better suggestions for you.
Also, is the "lawn" site in sun, shade, part, shade, etc.?
What is the soil type?
Generally speaking, I certainly would not recommend catmint for a yard you do not wish to mow--although in a droughty area, perhaps it would not grow as large as it does in my garden.
And are there any local ordinances about what may constitute a "lawn"? Some neighborhoods have rules about front yards.
Let us know what you did, torajima!

How is a lawn like this with dogs??

Clover and thyme are actually very resilient to animal urine.