Yes, I do it all the time.

Swimby, I meant to include a link to a picture frame where I painted the grout. On this one, I used copper metallic. What I do is seal the grout first, then paint with acrylic paints. I like to use a natural sea sponge (also called a silk sponge or a cosmetic sponge) to apply the paint. I wet the sponge first and squeeze dry, then squirt out some paint into a lid, then dab the sponge in the paint, then randomly sponge all over the mosaic. Then I wipe off the tesserae with a damp cloth. I like to use more than one color of paint. Just remember to wash your sponge often because the acrylic paint drys quickly. Then if you want a shiny finish, you can shellack the entire piece. I use Liquitex Gloss Varnish. I buy both the varnish and the acrylic paint at Michael's or Hobby Lobby.

Here is a link that might be useful: photo of painted grout