They dry quite fast for a natural, absorbent material, though obviously not as fast as a synthetic. They also feel drier much faster than cotton because the fiber structure is literally hollow tubes. So it sucks in the moisture and disperses it fairly rapidly, keeping the fabric dry, but cool to the touch. Cotton fiber structure for example is solid, flattish and ribbon like.
But since it has a high cellulose content (though lower than cotton), the Linen is not as insulative as say wool, silk, poly, or polypro. But due to the hollow, but also fine fibers, it's considerably more insulative than cotton.
Personally, i don't use it much for cold weather/winter backpacking. I switch over to wool, and wool synthetic blends for that for baselayers. I've wondered about trying a waxed Linen shirt for more outerwear for colder temps...