What is it about rayon dresses that the label often says “Dry clean only”? And why does it not apply to all rayon dresses? Just now I was doing my laundry and was starting to dunk one dress into the washer when I suddenly pulled up and remembered to check the label. Sure enough, it said “Dry clean.” I only got a little of it wet. Then I checked my other rayon dress and it said “Hand wash.” 
My WAG is that some clothing manufacturers are covering their butts by saying “Dry Clean” with some rayon fabrics. Also, they may not have preshrunk the fabric (it might not be economic to do so) so in order to avoid having the finished garment to shrink too much, they recommend dry cleaning. Yet, had they preshrunk the fabric (washed it before construction), the garment would have been suitable for washing.
Also, some rayon fabrics may take washing from someone who does it right (not too vigorous, cold water, mild soap), but since it’s too much of a hassle to instruct consumers on the right way to wash the rayon garment, they just say “Dry Clean Only.”
It is my experience that most rayon fabrics can be hand washed. I have made myself many (many many many—I’m ashamed to admit how many) rayon outfits, and I always ignored the fabric care instruction to “Dry Clean.” I hand washed the fabric in the sink, let it drip dry, and then proceeded to cut out my pattern and sew. Only a few times would I have been better off dry cleaning the fabric. (The one time that sticks out in my mind is that the fabric took on a more “cottony” and stiff texture after hand washing, losing the rayon “drapey” quality. I was still able to wear the garment, however, and no one else noticed or cared.)
Very occasionally rayon will shrink a whole lot (say, go from 45" in width to 36" or something like that), but most of the time the shrinkage was not too bad. And sometimes rayon will bleed out a lot of color when washed (not that I noticed that the fabric changed in color after washing). However, most of the rayon outfits I sewed seemed to last a long time, didn’t lose too much color, even though I hand washed them instead of dry cleaning.
Thanks for the very informative answer, yosemite! The hand wash sink in my laundry space hasn’t got the water pipes connected, so it’s useless. Anyway, my washing machine has a “Hand Wash” setting, the most easygoing level of mechanical wash, so I just use that and hope it’s good enough. It’s an even lower level of agitation than “Delicate.”
That sounds like just the setting if you are going to wash rayon. (Make sure to use Woolite or some other “delicate” soap on the rayon.)
However, you do run a risk of the dress shrinking. Like I mentioned before, some clothing manufacturers say “Dry Clean Only” because they know that if someone were to hand wash it, the garment would shrink and no longer fit. So, proceed at your own risk. (With that said, most of the time I’ve hand washed commercially made rayon clothes with no problem. But you never know!)
The label on my dress says “Jones New York. Made in Korea.” I wonder if a reputable label is more trustworthy in this regard than a no-name. I want to keep that desirable drapey rayon quality, the way the fabric flows and swirls, especially given the cut of this dress which was designed to enhance that very quality. What about fabric softener in the wash?
There’s no way (that I know of) to predict ahead of time which rayons will take handwashing well, and which won’t. And even those fabrics which take handwashing okay might still shrink (which is no problem when you wash the fabric before making the garment, but is a huge problem if you do it afterwards).
I personally would not hesitate to handwash a ready-to-wear rayon garment that was loose-fitting and not too expensive, no matter whether it said “Dry Clean Only” or not. But a really nice, well-designed dress? I would not take the chance.