As a former Vet Tech, I can honestly say that it's the most peaceful way to go. Generally, the critter will be sedated a bit with some diazepam, then the family gathers around and says goodbye as the pink liquid is injected into the bloodstream.

They really do just go to sleep.

I often think people should be allowed this dignity when we're ready to go. Instead, we're just left to linger till we're nothing but an empty shell. Sad really.