HI & Welcome...
If you are referring to COMPLETED download files than the question has absolutely nothing to do with LimeWire.
You simply must decide the most secure AND economic method to copy your existing files from your old comp and then copy them on to your new comp. Easy.
Certainly no reason at all to need to download anything again.
Easiest way (and fastest) is to use an External Hard Disk Drive... Just plug it into old comp via USB2 and copy to a folder on the external drive... Next, plug the external drive into the new comp and fill'er up !
Cannot give you specific instructions now cuz I've no idea what computer or Operating System you are using !
IMPORTANT: If you are using a CURRENTLY valid installation of LimeWire Pro, be warned that you must be careful in order to protect your license rights to your valid copy of LimeWire Pro...
They are NOT forgiving !
More info on this subject is available here in these Gnutella Forums...