After some practice, I came up with a pretty good technique. I'd start peeling with my thumb nail and then would get the whole circumference of bark peeling and would pull it at an angle perpendicular to the branch. This made for lots of long broad strips. (please excuse the crude illustrations produced from memory.)
The kids thought this was great fun and soon we had a small pile of bark strips. I took them up to the picnic table and started showing them how to weave them to make a place mat. They had done this for school with construction paper, so it wasn't anything new really, but using bark made it more interesting since the pieces were oddly shaped and had two colors, a green grey on one side and a bright white green on the other.
Weaving is pretty easy. Start with a cross. 
Then start alternating pieces above and below one axis, alternating axis each time to begin to give the structure some strength.
As you go, be sure to carry the under over pattern out so that you create a classic basket weave pattern. The more strips you add, the more strength the mat will have. To make it really strong, snug each new strip closely to its neighbors. This tightness will reduce play in the finish mat and help keep it together without needing glue.Eventually you'll have an odd shaped mesh. Trim it with kitchen shears to the shape you like. Here is ours, trimmed to the shape of an octagon. Also important, this is what it looks like after spending three weeks between the pages of one phone book, under a stack of 5 others. Pressing it as it dries keeps the bark from curling and helps form the wet freshly peeled strips to each other making a tighter weave.
To top it off, I stained ours with some Bombay Red polyurethane to give it that "fresh from Peer One" look. Here's the finished product. This is strictly a personal taste thing. It looks great green and would probably be good to go with a clear lacquer. I'm getting my office decor together as I am moving office as soon as the carpenter (me) has things tidied up ready for the carpet layer. And this is one of the colors I have picked for the wood work, so there you go.
You could probably get away with using this under a ceramic tea pot - a small one. This only measures about 4 1/2" in real life. The kids thought it was really neat and wanted to take it apart before I stained it. I think "stained" also now means "Dad's". :-)
