I was a Judo student in my younger days before I began Aikido.
At one Aikido Dojo we did the same excersise that Kev Price and Jun describes. That one seems to really work well.
O'Goshi was also mentioned which is a major hip throw in Judo. The Dojo I'm at now uses that for breakfall practice.
Your partner throws you with the hip throw very carefully and slowly so you get the feel of going over and landing. It has the advantage of the thrower being able to hold the throwee all the time. In O'Goshi nage has one arm wrapped around ukes back and holding uke close to them or grabbing their belt in the belt in the back. The other arm/hand is holding the sleeve of ukes arm and pulling it towards themselves to pull uke into them and unbalance uke.
You can pretty well force uke to fall correctly, hold them up and cushion their fall at the end.
We don't practice this regularly but it is a good training practice.
I also think prior to the actual fall from a throw practice that a class can do some fall positioning work. In Judo we would do excersises where you already were lying on the mat and practice the final position of being on your side with legs in the right position, arm properly slapping. We would roll left to right slapping the mat each time.
Mention was made of Judo by Jun. I had an experience not long ago where someone was knocked out by a Koshinage. I believe the person landed partly on their neck.
A comment or two was made that nage threw uke to low to the ground and that they should have lifted uke up with their hips to help give them clearance. In Judo we were trained to tuck our head under very quickly as nage would be trying very hard to throw you hard, fast and low to get the affect Jun mentioned where you don't so much lift uke as have him almost float off around your back. It feels very light but can be a pretty hard fall for uke.
I think in the training for Koshinage falls some thought might be given on emphasising this point of how your head gets under nage and tucks towards your own chest.