Hi Andy,
The stability issue has been mentioned by others. Here's a diagram to show how wood moves in drying:
To a lesser degree, the same cut of wood will react to changes in relative humidity throughout all the seasons of the life of the guitar. But, bear in mind, that diagram is sort of an average, and not all species will have the same movement.
If you want to know how any one particular wood will move seasonally, look up the radial (quartersawn) and tangential (flatsawn) shrinkage values.
Here's one place:
Wood Info (Look for figures representing wood at 6% to 8% MC, which is about where most of our wood will be when we work with it.)
For example, Australian Blackwood,
Acacia melanoxylon, is: radial 3.4%; tangential 9.0% That tells me that Aussie Blackwood is very stable quartersawn, but very unstable flatsawn. An extreme example: A guitar with a 16" wide back, built from flatsawn Blackwood in extreme humidity is taken to the desert, and the back shrinks 9%, which is 1.44". The guitar back cracks badly, in several places. OK, we don't build at high humidity or send guitars into the desert without humidifiers, but even 1/4 of that amount of movement would crack the back.
There are other woods, like Mesquite, with extremely low rates of shrinkage both radially and tangentially, as well as having the radial and tangential numbers in a ratio close to 1:1. Mesquite is bulletproof, when it comes to stability, whether completely flatsawn, completely quartersawn, or anywhere in between. Most woods are not. A very good example of a common guitar tonewood that is very stable is Cocobolo, at about 3% radial and 4.6% tangential shrinkage. So, if you like the look, you could build a stable guitar of flatsawn Cocobolo, and it would be about as stable as perfectly quartersawn Maple.
One wood that I try to remember is flatsawn Bigleaf Maple,
Acer macrophyllum, because it is sometimes available as gorgeous quilt figured wood, and a very well sliced set of quilted lumber is very close to perfectly flatsawn. Tangential shrinkage is 5.7% at 6%MC. There are plenty of quilted Maple guitars in service, that are holding up just fine, so I figure that any wood with a tangential shrinkage rate up to about 6% will be OK to use flatsawn. If higher than about 6%, the guitar better be a narrow body, like a parlor, and precautions taken to make sure the instrument does not suffer extreme humidity swings.
Remember also that many boards are not completely flatsawn. It takes a monster of a tree to cut 8" wide flatsawn boards. So, look at the grain, look at the shrinkage table, and sort of extrapolate for each board whether it is below 6% shrinkage. If yes, it should work, if not, keep looking.
An aesthetic factor: while perfectly quartersawn boards can be sliced to yield mirror-matched backs (and butts), flatsawn wood cannot. So, if you're a symmetry freak, like me, flatsawn wood may not trip your trigger. Just a personal choice, but if a back is pretty well quartersawn at the joint, then flares out to rift or even flatsawn, that satisfies my desire for symmetry.
Doug mentions another factor of using flatsawn wood: bendability issues. Not by my own experience, but from what I have read, flatsawn sides are much more likely to cup and ripple. That may not be true of the flatsawn material like Cocobolo, that is nice and stable even flatsawn.
Dennis