Crepe backgrounds have been working well for this bird series. I've created another crepe, this time for 14 shafts, and applied it as a jacquard ground with some 14-shaft granite weaves for accent, and a weft brocade for good measure. In this simulation, the birds are brocaded on alternate picks (they don't go from selvedge to selvedge).
There aren't enough woven samples to show you yet, but here's a simulation in Photoshop (click to enlarge):
If you're curious and want to try the 14-shaft crepe (you'll have two shafts left over to use for selvedges on a 16-shaft loom), here's a diagram, followed by a link to download the wif: