Quick update to the existing answers: the current version (4.3) of LibreOffice Draw opens .vsdx files indeed, although they appear a bit mangled so they'll require some work on it.
You could use an online service like http://www.freepdfconvert.com (there are dozens) to turn it into a PDF and with that view it or manhandle it in Inkscape, et al.
But as for direct opening, not yet, I'm afraid. It's fairly niche too so I wouldn't expect a solution any time soon.
Crossover handles Visio very well, unlike the many issues you may find with Wine. Although Crossover is a commercial product and Visio as well, as far as alternative Visio like applications there are a few.
I installed Visio in Ubuntu, but first I installed Wine. It's working fine (there are some issues!). I don't know if there is another way to open that kind of files. I tried with OpenOffice.org Drawing but nothing happen.
If you have access to Visio 2003 or higher, you could try to convert it to .SVG That format is easily readable by both Dia and Inkscape. If not, you're in a world of pain because raw Visio is not easily convertible.
With LibreOffice Draw 6.0.7.3 the *.vsdx Files from Microsoft Visio Standard 2016 that I have created there recently could be opened. Tested on Ubuntu 18.04.
As far as using LibreOffice goes, it's May 2018 and I'm using LibreOffice 6.0.1.1, and unfortunately LibreOffice Draw still won't open .vdx files as far as I can tell. :(
It's coming to LibreOffice now. Expect it in v3.5.
Quick update to the existing answers: the current version (4.3) of LibreOffice Draw opens
.vsdxfiles indeed, although they appear a bit mangled so they'll require some work on it.You could use an online service like http://www.freepdfconvert.com (there are dozens) to turn it into a PDF and with that view it or manhandle it in Inkscape, et al.
But as for direct opening, not yet, I'm afraid. It's fairly niche too so I wouldn't expect a solution any time soon.
Crossover handles Visio very well, unlike the many issues you may find with Wine. Although Crossover is a commercial product and Visio as well, as far as alternative Visio like applications there are a few.
Dia can open Visio's XML format (VDX)
I installed Visio in Ubuntu, but first I installed Wine. It's working fine (there are some issues!). I don't know if there is another way to open that kind of files. I tried with OpenOffice.org Drawing but nothing happen.
Open diagrams.net and drag and drop the file into the editor.
If the file is a .vsdx file and privacy is an issue use the desktop version
If you have access to Visio 2003 or higher, you could try to convert it to .SVG That format is easily readable by both Dia and Inkscape. If not, you're in a world of pain because raw Visio is not easily convertible.
With LibreOffice Draw 6.0.7.3 the *.vsdx Files from Microsoft Visio Standard 2016 that I have created there recently could be opened. Tested on Ubuntu 18.04.
Try the "Free Visio Viewer (Mac, Windows, Linux)" add-on in Chrome, by LucidChart. I just tried it in Ubuntu 14.04. It seems to work well.
As far as using LibreOffice goes, it's May 2018 and I'm using LibreOffice 6.0.1.1, and unfortunately LibreOffice Draw still won't open
.vdxfiles as far as I can tell. :(