Some people have commented that network share speed has decreased in the newer firmwares, which increases the chance of stuttering when playing HD media.
How are you connecting to your media: Wifi? Network cable? Directly attached storage?
If via wifi or network cable, try putting one of the HD files onto a USB flash drive or external HDD, plugging it into the Live and playing it that way. If it plays smoothly, you know it's network speed at fault, not the files.
Something else to consider is a 3-stage reset: that's a Factory reset, followed by resetting using the side button, then power cycling by pulling the plug for 10 mins.
The unit will reboot after the factory reset (Settings > System). Once it reboots, then perform the side-button reset. It'll reboot. Once it boots back up fully, pull the plug and wait.
Plug it back in and try playing the problematic media again.