

While it doesn't offer quite as much privacy as a "real" privacy perch, it could be a simple, fast, and free alternative to buying or building one.
Take a few old CDs, and cover both sides with scrapbook paper. I just taped it on. I already had the scrapbook paper onhand, but any paper would do. If you use scrapbook paper, one 12" square sheet will cover both sides of two CDs. Cut off any excess paper, including the paper covering the center hole of the CD.
Slide the CDs onto a dowel rod. I happened to grab up one of the dowel rod perches that came with my 30" cages, and the CDs fit onto it perfectly. I actually had to push/twist the dowel to get them on, but that's good because they can't slide around. The dowel measures 9/16".
Voila'!

The CDs are juuuust tall enough that the finches can't easily see over them while in a regular sitting/perching position. If they need to be taller, it would be easy to cut the paper a smidge larger in diameter than the CD.

When it eventually gets dirty, you could pull the paper off and recover the CDs, or just discard the whole thing and make a new one. It took less than 15 minutes starting from scratch.

(Technically you wouldn't even have to cover the CDs with paper if you didn't want to, but it looks nicer, I think. ;) )