I use one with a foam mattress that fits snuggly inside of it. You can buy a padded, quilted sheet though. I have one on my website under the "Travel" section if you want to see what they look like.

As for leaving your child unattended, what a weird warning! It is no less safe than a cot, there is no way my DS2 could climb out of it and I'm not sure what kind of trouble he could get into in there! It's probably just a disclaimer to protect themselves.