One option of compiling for Windows in Linux is via mingw. I found a very helpful tutorial here.
To install mingw32 on Debian based systems, run the following command:
sudo apt-get install mingw32
To compile your code, you can use something like:
i586-mingw32msvc-g++ -o myApp.exe myApp.cpp
You'll sometimes want to test the new Windows application directly in Linux. You can use wine for that, although you should always keep in mind that wine could have bugs. This means that you might not be sure that a bug is in wine, your program, or both, so only use wine for general testing.
To install wine, run:
sudo apt-get install wine