High Mass Walls
These walls systems are all solid walls made out of dense materials, of which the modern Brick, Block and poured concrete walls are the most common variety. Since these common variants are all have very low R-values, a fairly high embodied energy, and are familiar to most people, this discussion is about alternatives to those.
The common components of these wall are an aggregate of some sort (sand and possible stone), and a binder of some sort (Portland cement, Clay or a combination). In some of the wall systems, some of the aggregate is replaced with straw which gives the wall a higher R-value and increased lateral (side to side) strength.
The biggest issue with all these systems is that they don't produce very high R-values. As with most materials, the issue is that as density and strength go up, the R-value goes down, so in order to be strong enough to bear the roof load, these walls sacrifice strength. The walls with straw will have greater R-values than the ones without, all in proportion to how much straw is added.
All of these systems make very high mass walls, and as a result there are often claims of very high "effective" R-value due to the mass, but alas are not true for conditions that exist in the majority of the US (see "mass walls").
In order to increase R-values sometimes rigid board is added to improve the R-value.
All of these walls are at least somewhat vapor permeable, and most will absorb large quantities of water. All of the wall types made without Portland cement will disintegrate in the rain, so great care must be taken to protect the walls, lest they decompose before the desirable lifetime is over. The usual solution is to have large overhangs, although more rain tolerant skin like lime plaster can also be applied, or for that matter you could put cement stucco on, but since most natural builders are trying to avoid cement, that option is not taken. Most, but not all, high mass buildings without a cement stabilizer have been built in fairly dry climates.
Adobe is a mix of sand, clay and straw and formed into blocks which dry in the sun. It's a completely "natural" material that can be mad with low-skill labor from typically very local materials. The blocks are then erected into walls using a straw-clay mortar and are typically load bearing, but could be done in a post and beam structure as well. However, if you're going to do a post and beam, you should probably consider straw-clay instead, which has more straw in it, so has a higher R-value. R-values for adobe are hard to find, but appear to be in the range of .2 to .3 per inch.
Cob is nearly identical to adobe, but rather than being formed into blocks and then assembled into walls, cob is build into walls in place by hand, and as a result cob walls are highly irregular and often fanciful. Unlike adobe, cob walls don't have Portland cement in them.1
Rammed earth is a mix of sand, stone and clay compressed (rammed) into forms to form a dense strong wall that is similar to concrete. It is not unusual for Portland cement to also be used, especially in wetter climates, and this mix often goes by the name PISE. The only real difference between adobe and Rammed earth is the lack of straw and generally higher densities, which undoubtedly means a lower R-value. Rammed earth walls are compressed in "lifts" which leave a slightly wavy line in the finished product making it look a bit like sedimentary rock, a look that some find quite beautiful. In order to have a better R-value, some builders put rigid insulation in between two layers of rammed earth.
Autoclaved Aerated Concrete (AAC) - is a mixture of Portland cement and sand with additives that is formed into blocks in an oven, resulting in a solid panel filled with many tiny air bubbles. AAC has an R-value of about 1/inch, so is generally higher than the other high mass walls while still having pretty good compressive strength, although like the other alternatives here, AAC has a much lower compressive strength than the concrete, brick or block. AAC is made in a factory and hence has a much higher embodied energy than the other alternatives on this page.
Summary: Natural builders like adobe, cob and rammed earth due to its low environmental impact and that low-skilled labor can build them, plus they have a unique look that some like. The raw materials for all these wall are readily available, mostly cheap and will generally decompose with no toxic residue. Since high mass walls don't function well in most climates, and are outperformed by other wall systems, even in the high, dry climates they work in, they are only used when there is some other reason. Combined with a suitable skin of insulation, they could perform quite well, although with the advantages and limitations of any high mass interior.
Notes
1: Of course, saying that means that someone, somewhere probably did put Portland cement in a cob building, but I've never seen it documented yet its easy to find cases where adobe has added cement.