Visio is not developed for the Mac OS X or Linux operating systems; other developers offer diagramming programs for those systems. Due to proprietary Visio file format, few Mac OS X or Linux programs can read Visio files. Omnigraffe Pro on the Mac can read and write Visio files.
That said there are a few alternatives out there that will run open source on a Linux platform. I can't say how good they are. Open office Draw which is mentioned as an alternative is not quite Visio. However you can check this out and see if anything listed as an alternative is any good.
Gone through the list and happy to say I have used some of them during my software engineering days. If you are looking for pure open source then Dia is the best, all the it is not very user friendly and functional. But still there are open source fanatics out there. Diagram.ly is unique in the sense you don't have to be logged in to draw diagrams. But saving and coming back again to do work it very hard with them since file export and import is needed every time. Also they have very few object libraries. At the moment I'm using Creately as my diagramming tool. They have a page explaining why they are a good Visio alternative. They are flash based so not every ones cup of tea, but still a very good cheap alternative to Visio.
I use Visio at work and love it. I wanted something similar at home and was recommended Dia. I also recommend Dia as it's easy to use after about 10 minutes of playing with and importantly for me, it imports jpegs and exports the final doc to PDFs.
The multiple pages on the screen can be annoying, but hey - It's free.