What files do I need to copy to a new computer to prevent having to sync with the network? I think this used to be the blk0001.dat file, but it seems that the block chain is now (since version ?) moved to the blocks/ folder, containing a lot of blk* files, as well as an index file and multiple rev* files.

What files do I have to copy to a new machine if I want it to be synced up?

I know I will have to run with -rescan the first time after I changed the block chain files. And I know that the blocks are not verified (but they come from another machine I own, so there are no trust issues).

Please mention the current version number in your answer to not confuse people if the file configuration would change over time.
https://en.bitcoin.it/wiki/Data_directory

This might be of some help.I followed that and successfully moved the files from C:\Users[username]\AppData\Roaming\Bitcoin\ to E:\BitcoinData.

Just cut and paste the whole content of from the Bitcoin folder (C:\Users[username]\AppData\Roaming\Bitcoin) and set the new target as 

"C:\Program Files (x86)\Bitcoin\bitcoin-qt.exe" -datadir=d:\BitCoinData and that is assuming you have bitcoin Qt installed in your C: drive.Change the path accordingly.