I fixed my problem, but not the way anyone suggested here. My problem was basic.
New items on a list would not show up on the device unless I created an item in the list on the device. Then it wuold 'sync'.
This has nothing to do with Background Refreshes in iOS 7 (Reminders - on my devices - isn't even listed there). If you are using an iCloud account for your reminders, this is totally related to:
Settings > Mail, Contacts, Calendars > Fetch New Data > Push
AND
Settings > Mail, Contacts, Calendars > Fetch New Data > iCloud (your specific account) Push must be enabled
If you have the master push button on and you have push setup for the account it works fine.
Give it a try.