Just spent a few hours figuring out the right concoction of ffmpeg arguments for transcoding my collection of digital videos for playback on our new 2014 QX60. This results in full screen (i.e. no black bars) playback, at full res. I've thrown MKVs, MP4s, AVIs (DivX, Xvid), and about 18 movies so far at it and all have resulted in perfect support.
ffmpeg -i <input file name> -y -threads 2 -vtag DIVX -f avi -vcodec mpeg4 -b:v 1071000 -aspect 16:9 -s 720x480 -r ntsc-film -g 240 -qmin 2 -qmax 9 -acodec mp3 -ab 128000 -ar 48000 -ac 2 -benchmark <output file name>
All the software I could find to do it either resulted in black bars, was Windows only (I use a Mac, though the above ffmpeg command should work on any supported platform: Windows, Linux, BSD, etc. assuming ffmpeg was compiled w/ the right codec support), or cost $$$ (which I'm cheap and didn't feel like spending).