Yep! Go for it. You won't encounter any problems.
You can just copy the entire cygwin directory to your new machine, open up the cygwin shell and everything (as long as you are only calling cygwin-internal programs and stuff that's within the path) will just work as if you you are working on your old machine.
The only thing you'll loose is the directory where the "already downloaded and compressed" packages for a possible re-installation are stored. Fortunately this directory is optional, so no problem for migration to another platform. You could copy that directory as well, but most likely all the packages that you have are outdated anyways and a run of setup.exe would fetch the new versions anyway...
Btw - since someone said exactly the opposite some real-life experience: I use this feature quite often with success. I've copied my cygwin dir to USB-sticks and used it on friends computers. I also copied it to the laptop of my fiance when we go to holidays and take a laptop with us.
It always worked without any problems....