Answer: Use the AirDroid app.
Investigation:
I'm also using Droid Turbo with Linux Mint 17, and I had the same issue.
One thing worth trying is sudo mtp-detect:
$ sudo mtp-detect
Unable to open ~/.mtpz-data for reading, MTPZ disabled.libmtp version: >1.1.6
Listing raw device(s)
Device 0 (VID=22b8 and PID=2ea8) is UNKNOWN.
Please report this VID/PID and the device model to the libmtp development team
Found 1 device(s):
22b8:2ea8 @ bus 3, dev 11
Attempting to connect device(s)
Android device detected, assigning default bug flags
USB low-level info:
...
Then if you do sudo mtp-connect, you'll see this:
Usage: connect
Commands: --delete [filename]
--sendfile [source] [destination]
--sendtrack [source] [destination]
--getfile [source] [destination]
--newfolder [foldername]
I found that using sending a file to the root path / works, and it shows up in the SD Card folder on AirDroid. Trying /SD\ Card or /sdcard didn't work.
Here's a working example of sending a ringtone file to the correct folder:
sudo mtp-connect --sendfile Moonlight\ Sonata\ Ringtone.mp3 /Ringtones
It's a bit slow, but it works. (Note: Don't put a / after the destination folder)
Here are some useful folders:
/DCIM/Camera
/Notifications
/Music
But seeing as the AirDroid application works well for all of this and makes seeing the file structure easy, that seems like the best solution.
Disadvantage of AirDroid:
The file has to pass through the Internet, so it could be slow.