Will Linux run my stuff
Quick answer yes and no. Linux is as different as Mac and Windows9x applications. However there is a group developing a project called Wine. The goal of this is to provide a win-32 compatible set of libraries which are hooked into Linux and X-windows. This is a total re-invention of code based on observered behavior of the comercial stuff. Currently many applications run on Wine.

Disclaimer
There are many published bugs and incomplete develpment warnings on the web site. Be warned, things may or may not work as expected. Personally, I haven't tried it yet because I run all native Linux/Gnome applications. I haven't found a need to run Wine.

Freedos and dosemu
These are two separate but somewhat coordinated projects. Dosemu is a Linux application that runs to provide a dos compatible emmulation. Basically it provides an interface layer between Linux and dos to make it look like dos is running on a dedicated computer. Freedos is a msdos equivalent. It reportedly runs any dos application with a few catigorical exceptions of direct hardware resource usage.

Bigger disclaimer
I have tried this combination and had my computer lock up. Yes, it locked up Linux. At the time I had a also fan problem which caused hardware related crashes, so its possible that was the real cause. I also couldn't get floppy drive working, but I figured out a work around sort of. I plan to try it again some time. This would be useful since there is a ton of dos stuff for ham radio. Although I would probably look at how the dos version worked and port it to Linux if I found something useful.