Upvote:3
I did transport a desktop computer over an international flight last year. I did not travel throught the United States however (through Canada), so I'm not sure about what the TSA may or may not do. I ended up bubble wrapping it and putting it in my suitcase that I checked in. The cushioning is important. As long as it is cushioned, it should be fine.
Just as an extra tip, you can ask the person at the checkin counter to mark the suitcase as fragile. Don't depend on soft handling though. Pack for rough handling.
If you end up taking components out, you will have to make sure you wrap them in anti-static bags and handle them carefully all throughout your travel since I'm assuming you'll take it as carry on.
EDIT: Might be wise to take out the hard drive if you are using a HDD instead of an SSD.