On a 4th Generation 8GB Fire HD which had no way to mount a SD card I used a San Disk DuelUSB Drive plugged into the usb/charging socket. The software on the tablet had been updated to version 5.1. Going to the video app on the home screen it was able to find a list of movie files on the Duel Drive and play the movie file with no problem. Don't forget to have the files you want to play in a standard format so they can be read by the software on the tablet.
Of course there is a risk of damage to the USB slot with a 2cm drive sticking out of the side of the tablet so I only leave the drive in place when i want to play something on the drive and thats not that often. Its also important to also eject the USB drive correctly using the command to eject before removing it rather than just removing the drive by yanking it out so the file being used isn't corrupted