First thing I would do is update your GPU (graphics card) drivers to the latest version if they haven't been already. You'll be able to find them on the AMD website - those Blu-Rays might need more recent drivers to work in the case of an HDCP update.
Another possibility is that the newer Blu-Rays are using a newer type of copy protection that your BD-R doesn't know how to decode. In all likelihood, the BD-R drive's manufacturer has made new firmware for the drive so that it can decode the newer protection type.
Install the latest firmware for your Blu-Ray drive. Google
'[your blu-ray drive's model and manufacturer] firmware'
The only other solutions I can think of are workarounds such as using http://www.slysoft.com/en/anydvdhd.html
, but we're not allowed to help you with that.
AFAIK, your graphic card has nothing to do with your Bluray drive. The drive itself sends video to the graphic card. The graphic card just projects the video.
Not quite true - you need an HDCP-compatible GPU to output any video+audio from a BD-R. So it's quite possible that a driver needs updated somewhere.
Are there different grades of Bluray discs?
There are different regions, and single/dual/triple/etc. layer discs, though it's really just the region that makes a difference.