Cleaning Sticky Painted Wood Carvings?

I would use Murphys Oil Soap

Dust first and then gently wipe with damp cloth; dry immediately.
If these are ornate carvings you can try using canned air (generally used on computers) as this can blow a lot of gunk out of the crevices.

You can then try mixing about a cap-full of Murphy's Oil Soap in a quart or more of water and wring out a clean cloth and gently wipe the surface.
You can dip a toothbrush in the mixture to clean out the crevices.
Always rinse cloth/toothbrush in a bucket/pan of clean water before reusing.

Use clear, clean water as final wash.