My reading indicates that linenfold panels were developed in the second half of the fifteenth century. Panel frame construction seems to have really gotten going in the late fourteenth century, and most panels hat flat fronts and a thicker center on the back, making them kind of triangular in cross section. But flat fronts are boring, in the Middle Ages displaying your wealth was the name of the game. From here they could go a few different ways: Gothic tracery could be nailed to the plain panels; they could hire an artist and paint the panels; they could do both (tracery framing a painted scene), or you could simply carve the panel itself. At some point, somebody flipped the triangular panel around and put the thicker part on the outside, and then carved the top and bottom edges to look like a layer of cloth or paper. This was probably the first linenfold pattern, which was pretty simple.
By the start of the 16th century, linenfold became much more complex. Multiple overlaid loops of "fabric" were created. Some even had carved "trim" on them. By the mid to late 16th century, linenfold became much more stylized and abstract, with "holes" in the layers, less plausible geometry on the folds, lots of straight lines without looping transitions, etc. I don't like this abstract style so I haven’t done any real research on it.
OK, on to the work. This is a very small, simple panel, only 9-1/2" long and 4-1/2" wide. There is a 1/2" rabbet around the carved surface to allow it to fit into the frame, so the overall carved surface is only 8-1/2 x 3-1/2". I call this a two-layer carving since there is a most two layers of overlap. Here’s the pattern:
All period linenfold that I've seen is symmetric. That doesn't mean you have to do it that way, but it does make things easier. The carved areas on each end are about 3/4" high. The panel itself is only 1/2" thick and the carving thickness will only be 1/4", You can go much thicker if you want, but this works pretty well.
The panel will have two hollows adjacent to a center rib. Each hollow is made by a round bottom plane. You can clamp on a fence to start that plane, but I find it easier to use my plow plane to cut a groove since the plow has a fence I can set to the correct spot.
Once the groove is cut, it will guide the round plane as long as you plane lightly.
Here the round plane has cut the hollow. This isn't strictly round, since my groove was a little away from my central rib. You can tilt a round plane (and adjust the iron too) to make it cut more to one side. In this case I cut toward the center after I went deep enough to remove the guide groove.
The next step was to round off the outer portion of this side with (in my case) a skew block plane and a #4 smoother. Once that was done I redrew the pattern that I'd planed away. Symmetry helps here, since the other side of the panel still had it's pattern.
Panel flipped around and now the second hollow is cut, forming the central rib.
Remainder rounded off.
Here's my round plane's underside. You can see the cutting surface of the iron is only about 3/4" wide.
Starting to chop one end. I usually cut away in layers, starting with the wood that will be at the lowest (background) layer. A straight cut down with a gouge outlines the cut. Note: I do the vast majority of my carving with a mallet, particularly on oak. You would be at it a very long time if you're just pushing the gouge into the wood by hand.
Here the ground section has been removed. After the vertical cut, I usually pop off the waste by chopping in from the end grain. For deep cuts like these you'll need to pop them off in layers of up to 1/8" deep. In difficult grain it you have to make sure the grain isn't diving down too deep so you remove too much.
Now to start on the end folds. I'll usually dig out the curve with small deep gouge to start
Here's that cut of the gouge. This tends to leave a "realistic" edge.
After the edge, I repeat the chop down/pop off process to complete the removal. Note that I also undercut the top layer to give a more 3-D effect.
Here's the end after cleanup. I do a slight chamfer or bevel on the carved edges. This catches the light better and also reduces accidental snags when the panel is in place. Note: I will go over this with a fine wire brush to remove the semi-connected splinters. But once there is finish on this more defects will show up and it'll get some more cleanup.
Next I flipped the panel around and did the other end. You will note small differences between this end and the other. This is where hand work is different from, say, CNC. You can make alterations for grain, or if a chunk breaks, or there is a knot or rough spot.
More complex carvings can be made by adding more folds and more cutaway layers:
Total time on this panel, from bare wood to completed carving, was about an hour. Note this is fairly small, a simple pattern, and this was panel #16 of an 18 panel project so I'm in practice. You might need more time.

I used 4 gouges (three with shallow sweeps and a variety of widths, one with a very curved profile), one chisel, and five planes (plow, round, #4, #92 rabbet, and a skew block plane) on this project. You could get away with less.
Kirk