I have a directory called downloads I use for torrents, but it seems to have become "infected", although it's quite possibly a windows virus.

Ever directory and subdirectory has been filled up with a large number of odd files, identically copied to each folder below the root downloads folder

< partial snip of ls>
Norton Antivirus 2005 beta.exe
Opera 11.exe
Partitionsmagic 10 beta.exe
Porno Screensaver britney.scr
RFC compilation.doc.exe
Ringtones.doc.exe
Ringtones.mp3.exe
Saddam Hussein.jpg.exe
Screensaver2.scr
Serials edition.txt.exe
Smashing the stack full.rtf.exe
Star Office 9.exe
The Sims 4 beta.exe
</end>

doesn't appear to have done any damage though. Anyone know how I can remove the files in one fell swoop.

I think that you have to find out what is creating the files. I am not familiar with bittorrents software but I would first look to see if it is running downloads repeatedly.

Now this file looks suspicious:

Saddam Hussein.jpg.exe

However it could only cause a problem for you if it has Java code or if it has a Windows virus that you allow to run by using a Windows file viewer through Wine. Nevertheless it is unwise to download such obvious sources of viruses even if they can't run on Linux. Security starts between your ears.