In reading Hedwig's post and talking to other fellow burpeers having low back problems I thought I would share some advice.
I have been combining this challenge with lots of yoga - good for the low back and stretching out all the muscles that the burpee uses. Also good practice for push-up form.
If you don't have time to do full yoga try at least 4 cycles of the sun salutation series: standing forward bend, jump back to plank, push-up position, upward dog, downward dog, up to standing. Take your time if you are tight.
Here is a video explaining how to do it. In two parts