How about constructing raised bed boxes from 2by 8 or 10's and then bracing them inside with extra wood so that they will keep their shape. By bracing, I mean having another piece of wood running through the center to hold the shape. Should work.

You have to build a retaining wall out of stone to get higher than 10-12".

Small beds of one foot height or less can use the standard garden centers' Retaining Wall Blocks. Pick your style and color. ...I suggest a small bed, very small, like 2'x4' x 1' high as an initial project. It is not hard but is labor intensive. You will need to get an inch or two into the subsurface...

4"x6" lumber will work for that size too; even 4x4's. However, you will want to stack heavy stones on them so 4x6 is best(with the 6" side on the ground). You can use cedar or redwood if pressure treated cannot be used. ( BTW, That is expensive lumber. )

For taller structures you need some larger blocks such as those from Anchor Wall Systems. They are called gravity walls because gravity is the equalizing force to sub' for those stakes. You can build 18", 24" and for some of the blocks even 36" gravity walls.

Anchor Projects
AWS Products

Now if you have big, heavy(50+lbs) stones in the area, you could use those rather than purchase cut ones. Gathering them should be fun -take your shotgun. ...
You can make a frame of 2x12xY' lumber and stack stones on and around it too. The frame by itself will not work without support in the long term. (short term w/b okay)

I rec' that you look for landscaper in the area and tell them what you want. So many beds is a big project and take my word for it, those retaining wall blocks used for 2'+ high walls are very heavy(70 lbs ea.) when moving and stacking a hundred or more of them. And you will need a lot more than 100 ...

Now, here's a novel idea for you if you want to be earthy and practical. Use bales of hay. Entire homes have been built that way. I have no experience building structures with them so you have to Google...

Just so you know, pressure treated lumber is not harmful to people, plants or the environment when used properly. Five years or so ago, it might have been ( and was dangerous to some really stupid people ) but that kind of PT lumber is not made anymore or sold anywhere in the USA.
It would be, relatively, simple to construct a 8'x8'x1' bed with landscape timber; I've done it here and am about to do it again for a base wall for a garden house. It is like making toothpick houses but instead of glue at the corners one uses 5" spikes ... there are plans on the net. E.g.: Raised Bed Project
Again, however, without stakes in the ground you will have to add weight to the top. The taller it is, the more that is needed. Sides will need support too if the structure is very long...

Good Luck!

violet_z6 posted a bunch of great pics here:
Raised Bed Insulation (Follow-Up #8)

With some creativity you might get around having to build stone walls to get depth greater than 12".

Here's a pic' of "dry-stacked" flagstone beds to get an idea of what you can do:

I live in the same kind of area...The Texas Hill Country has a few pockets where you can find dirt but mostly rock...the answer is raised beds.....We used landscape timbers...yes I know some people object to them but I chose to use them...and we used spikes from Lowes or Home Depot and anchored the three together...they are layed out so that they overlap each other including on the ends and I promise you...these beds are not going anywhere....the spikes are like huge nails and we pre-drilled to drive them into the timbers.

We have an old walkway beside the house we wanted made into a garden space. We dug up some other walkway areas and pounded the cement with sledge hammers. I used the small junks as the drainage mechanism over the solid cement walkway and then used the old walkway for "stacked stone" - I did this right over the stairs too. It looks great and nobody ever guesses it was a walkway, or that the cement is not real stone!

GGG

Here is a thought, you could look around and get old bath tubs, and skirt them in with wood, or rock I suppose. I've seen these done at a seniors center and they looked great, couldn't actually tell what they were. Would give you more depth than 18 inches as well.

You could use a zillion different things. White wash some old tractor tires and set them directly onto the rock. You'll want to whitewash them so they don't get heat up the soil too much in hot weather. I love the stacked stones. We used to have a raised bed garden directly on an old cement slab patio in our backyard, and I just used old boards to build a 4x4 box and ran a few extra pieces acros the bottom for additional support. I didn't grow carrots or potatoes in that one, but it was great for growing shallow-rooted veggies.

girlgroupgirl - do you have a picture of the walkway garden you describe? I can't quite imagine it, but it sounds interesting.