Over the last year and a half of making prototypes I've colored wooden bits a wide variety of ways. I've painted them with acrylic paint, I've spraypainted them, I've used wood dye, and I even nearly tried fabric dye at a couple of people's recommendation.
One failed method involved just trying to color them with permanent markers -- as with most markers when you go over the same place twice it gets much darker and so an even color is very difficult.
The other day on a whim I tried dry erase markers (Expo brand), with excellent results. The colors are very even, rich, and actually dye the wood (so that you can sand them a bit afterwards if you want to). A quick coat of spray acrylic on top et voila, beautifully colored wood bits.
I'm quite pleased with the results!
When I said "wood dye" I should have said "wood stain," sorry. Wood stain is the traditional coloring they use to color wood, usually available in a wide variety of shades of brown but with occasional other colors like green and red and blue. It's very messy to apply. I bought some wood stain pens at a craft store and they worked ok but were very expensive and the color selection was very limited (they have similar pens at woodworking stores, too).
I use Krylon Clear Coat brand spray acrylic.
I don't have a supplier for meeples, unfortunately. I use all kinds of other wood bits, though, from craft stores (like Michael's) and a couple of the wooden bit supply places listed in the Resources/Links section here.