You can install ImageMagick and use the identify tool with -verbose, for example:
for f in *.jpg *.jpeg *.JPG *.JPEG *.mp4 *.MP4; do
identify -verbose "$f"|grep exif:DateTime
exif:DateTime: 2013:04:08 10:45:44
exif:DateTime: 2013:11:01 22:59:27
exif:DateTimeDigitized: 2013:10:31 23:39:55
exif:DateTimeOriginal: 2013:10:31 23:39:55
exif:DateTime: 2014:07:14 10:48:43
exif:DateTimeDigitized: 2014:07:14 10:48:43
exif:DateTimeOriginal: 2014:07:14 10:48:43
Sorry this is not a complete solution, but if you're familiar with a scripting language like bash or Python, you could easily write a script to parse this output and decide which folder to put each one in.
As you can see some files have 3 DateTime fields, some have none, so you may be left with a few that need to be handled manually, or using the file modification time as fallback.