I'm currently deprived from any Windows machine, and Mono doesn't run my application. Is there another way that might get around any reason why Mono wouldn't work? (I don't have the error message at the moment.)
Other than virtualization, no. If you have the hardware to do it I would recommend VMware.
You could try running the .NET Framework under Wine. I wouldn't expect much out of that, and the only version that has gotten the Gold rating was 2.0.
If you don't have the cash for VMWare (if you do, it's the best money can buy), try VirtualBox. It's free, open source and it's really good. Not as powerful as VMWare though.
*rolls eyes * and this is why I don't recommend using .NET as the right tool for ANY job. It's not cross platform compatible, no matter what Microrip-off say.
Back to business. Look into CodeWeavers Wine. They do a lot of stuff to make their package of Wine more compatible. A lot of the work is in settings to run each app, but they do also patch it quite heavily and those patches can take a little while to work their way back to mainstream (they do give them back though).
If Virtualisation is what you go for VirtualBox is pretty good. I use it on the Mac and it does everything I need with reasonable speed. Parallels seemed a little faster, but if I didn't reboot my Mac for a long time Parallels seemed to run like a dog after a while. VMWare offers roughly the same features/performance as parallels. Parallels feels a little better in terms of its Mac integration but it's all personal taste.
If you need Linux compatibility as well then look at VirtualBox or VMWare. Both have free versions for Linux (VMWare Server is available at no cost). If you go with Virtual Box or VMWare on both Mac and Linux then you will be able to pass around the same VM files rather than having a Windows VM to run in Mac and a Windows VM to run in Linux (not to mention the licensing costs of having two copies of windows installed).