There aren't any true replacements for Visio as the file format Microsoft uses is very proprietary, however there are several other programs that can create flowcharts and other vector drawing. The best one I've found so far is Diag. I've been able to create flowcharts and diagrams with it quite easily. There are several more , but I've never used them. So if your wanting to create new flow charts etc then you can use Diag, however there is no Linux program I know of that is capable of opening existing Visio files.
Anytime you want to find a good alternative to a pay Windows program you can check out this website. www.osalt.com
It usually has what I'm looking for and provides several alternatives to popular pay software like Visio.