Look for wooden sunglasses with some sort of UV protection. If it has any, it will have a sticker saying what kind of protection and how much. UVA and UVB combined protection is ideal. If you're not sure the pair you purchased has the protection it claims, bring it into your local optometrist. They'll be able to tell.
Another thing to consider is the coating on the lenses. We suggest a polarized pair for all around safety and protection. Polarized lenses minimize glare and bright reflections from surfaces such as water, windows, asphalt roads, and snow. Polarized lenses are pretty common on sunglasses these days so it shouldn't be a problem to find them on a wood pair as well.
If you have specific interests or hobbies that keep you out in the sun for long periods of time, you may want to consider finding wooden frames with colored lenses. There are plenty of different colors and they all benefit the wearer in different situations. A general rule of thumb is that dark is great for everyday activities. Darker shades will cut glare and minimize eyestrain for moderate to bright lighting conditions. Note that gray and green will have no effect on color distortion, while brown may. On the other hand, light tints are better for low to moderate lighting conditions, such as all snow activities. Lighter lenses allow for better depth perception and brightening up surroundings.