Gravity alone might not allow more dense stuffed animals to drain properly; squeeze the water down from the legs and out through the feet (or whatever...
"Surface Wash Only" is a myth. Standard stuffed animal toys can be submerged and washed carefully. I'll show you how to rescue your most loveable (and roughly loved) from being discarded on the grounds of being "unsanitary."
Find one that really needs a bath. This guy has a mystery stain. Gross.
Step 2: Submerge!
Soak in cold water, submerging and squeezing. Nasty stuff will most likely come out of the stuffing. Keep squeezing it out until the water comes out clear.
Step 3: Scrub
Scrub your critter with a mild detergent (i make my own with Ivory soap). Cold-water liquids like Woolite work well, but your usual liquid detergent or liquid dish soap are good too. Use a toothbrush to really get down into the fur. (sorry, no photo of the scrubbing action; can't do that one-handed)
Step 4: Squeeze
Press out the excess soap by squeezing your animal under the water again. You'll probably see more gross stuff. Continue until the water coming out runs clear.
Step 5: Rinse
Rinse under cold water and continue squeezing out the suds.
Step 6: Hang to Dry
Hang outside on a line, because these guys drip a LOT. If you can't put them out in the sun, devise a way to hang them over the bathtub or other area where water isn't a problem.
Step 7: (optional) Squeeze Again
Gravity alone might not allow more dense stuffed animals to drain properly; squeeze the water down from the legs and out through the feet (or whatever is hanging lowest from the line). The motion is admittedly not unlike milking a cow.