I understand there is no converter better than LibreOffice itself.
I recently read a feature on this issue in Linux Format magazine, and this is what I remember of their advice:
1. Make sure you have the latest version of LibreOffice
2. Make sure you have the free windows fonts installed in Linux, so the fonts will be exactly the same when you open the document in word.
3. Install a free DOCX viewer from Microsoft in Wine so you can preview your documents and make sure they are good.
4. Install a version of Word in wine.
I left the magazine in my office - I will bring it home tomorrow and review it again, to see if there is anything important to add.
PS. The other important recommendation was to use Microsoft Web Apps to check and fix your documents formating before sharing it. This works well with Firefox and Chrome, but apparently not so well with other Linux browsers.
Notebook: HP 14 Notebook PC 14-r003TX, i5-4210U @ 1.7 GHz, 8GB RAM, Linux Mint 17.2 (Dual boot with Win 10)
Notebook: DELL latitude E5520, i5-2520M @ 2.50 GHz, 4GB RAM, Linux Mint 17.2