Marie Noelle Fequiere

Welcome to Aikido, David. The point to proper falling is actually to protect your back, and this knowledge can be extremely useful off the mat. I've heard people telling how good ukemi helped them when they tripped in stairs, or even fell off a horse.
I too have a bad back. Years before I started Aikido, I took a nasty fall, and even though I never went to a doctor, I believe that I had a fractured coccys.
Ukemi is not a problem for me, but, strangely enough, meditation is. Sensei sometimes have lie flat on our back to meditate, and I cannot tolerate this position for more than a few seconds before my lower back starts killing me and I have to bend my legs with my feet flat on the floor, wich makes relaxing my legs a challenge. Sensei has this ki exercise that he loves, when we have to imagine that our body is like a piece or steel. I notice that kids are just great at this exercise. They can be lifted from the floor and placed with their head on one chair, and their feet on another one, and another kid stands on their stomach, and they don't budge. This stunt is not for me, but it's not an exam requirement, so I'm fine.
Another problem is irimi nage practice. After having my partner bend me backward a few dozens of times, I have to ask permission to step to the side and stretch my back.
I also do stretching and strengthning exercises at home.
Just tell your instructor about your problem, but I promise you that it won't keep you from learning Aikido.
So again, welcome to the club, and have fun!