Brick garden structures without concrete footing: am I crazy?

Every “intro to raised beds” article mentions that they can be made of many materials, including brick, but there’s not much online about how to actually build them. Every tutorial I’ve found starts with “pour a 12” concrete footer.” I don’t want to pour concrete into my garden, I just want a durable raised bed that matches the brick work in my house! I live in a mild temperate climate with an official frost depth of 12”.

Would it work to build a foot-high brick wall over a footing of gravel? Or should I give up and accept replacing my garden beds every 4 years?

I've built block walls by digging a trench one course deep,compacting the soil in the trench, and building atop of that. Level the first course with gravel,sand,or stone.
No settling,over 10 years.
Your gravel filled trench should be fine, though a single layer thick wall might be unstable.
The bricks with the holes in them could be stabilized by driving rebar  through them.

Trying to keep the bricks level as you're building may be more of a chore than the footing, since it gives a solid/level starting point.

12" deep footing may be excessive. In my area 4" is adequate.

You could make a brick footing, it's just more expensive and a lot more work.

The point of a footing is to have something solid that extends down below the frost line so that frost heave doesn't break your wall.  Concrete is used mostly because it is (relatively)cheap, reliable, simple, conforms to the surface below it, and forms a single, monolithic, footing.

With an 8b zone, I doubt you'll have many issues with frost heaving. The reason for most footings are to support the weight of a structure, but with 1' high walls, no floors, ceiling, furniture, etc you'll be more than fine. I'd be more concerned about lateral pressure forcing the bricks outward. Something like William mentioned with the rebar should do just fine. Other options might be some kind of lateral strap around the structure (like a cable or steel strap), a thicker wall, or some kind of deadman device.

All in all I wouldn't worry much — this isn't a structure you're gonna sleep in. If it collapses, it's not going to hurt anyone.

Whatever you decide I'd recommend a self-watering wicking bed system for your raised bed. In Australia the days get very hot in summer and whilst my regular garden bed is completely barren my wicking bed containers have thrived all summer.

It would mean the real structure is just a thick piece of plastic holding everything together, with brick or wood plants being a frame for it to cling on to.

In my opinion, if your raised beds are only going to be 1' high then you don't really need any retention. A simple berm or mound should do the trick. Going to the trouble of digging any type of footing for a 1' tall retaining wall seems like a lot of wasted energy.

My personal aesthetic is much more loose so in my gardens a lot of the edging is accomplished by rip rap or logs from the woods. Either that or no edging at all. Is there a reason other than aesthetics which drives you towards a brick retaining wall?

As mentioned we do not have a problem with frost heave in the maritime area at lower elevations. You can make a foundation by digging the soil out and leveling the subsoil and lay the first layer of cement blocks on their side. If you chose to make the wicking bed mentioned above [these work very well for me] dig the whole bed area to the same level then after placing the foundation blocks put down a liner that extends over the foundation blocks. This will give you depression to place 3 inch diameter perforated drain in. They come in 10 foot lengths so to make a 4 foot bed buy one length plus 2 elbows. Cut it in half and then a little more than a foot off of each one to make the riser to add water. place the wall on top of the liner so that excess water can drain out under the bottom of the wall. Fill around the drain tile with sandy gravely subsoil that will be filled with water them put the good soil to the top of the bed. Fill the tubes with water and put a water bottle in the riser with a stick in the lid as a flag when the water level drops too low. The whole soil column needs to be damp to maintain the wicking action so if the soil dries  out water from the top again to restart it.