Greg: Don't use RIT dyes.
I did use black RIT dye on some other grilles, and they looked fine for a couple of weeks, but then they turned brown. I also had the problem that the plastic-y frameworks of the grilles didn't like to be boiled in the dye solution -- they got distorted.
I'll look for cold water dyes -- thanks for the suggestion. I also ordered some black dye that is specially for polyester, since that was my first guess about the material of the Axiom grilles (but it has to be applied hot).