Is there a way to delete a line in Visual Studio without cutting it?

I mapped Ctrl + L (Global) to Edit.LineDelete. Otherwise, the shortcut key is Ctrl + Shift + L, which is awkward. Go to Tools > Options > Environment > Keyboard as shown below.

Ctrl + Shift + L will delete the line and not copy to the clipboard.

Correction in my answer

Ubuntu 16 & Visual studio Version: 1.30.1

To cut line Shift + del

To delete line Shift + Ctrl + k

Edit.LineDelete is the name of the command. By default it's bound to Ctrl + Shift + L, but you can give it whatever you like in Tools | Options | Keyboard.

Edit: Corrected default shortcut info.