A polishing compound called "Zam" is the right stuff for polishing turquoise on a muslin buff wheel, either on a Dremel or a larger buffer. A hard felt wheel is too hard for Zam polishing, it works better with Tripoli for initial smoothing.
You can probably find Zam at a local rock hound or lapidary supply store. We carry it at Rings & Things as item # 69-187
. It's really made for polishing silver set with turquoise.
If it is natural turquoise the ammonia in some cleaners may dull the stone. Zam should bring the luster back. If the chunks have enhanced black dye in the crevices to look like matrix the Zam may polish it out.