I had a similar problem a few years back in my old raised bed garden.
You were able to see the mole (or vole) tracks just beneath the surface of the grass, leading towards the garden beds.
I remember looking at the garden in the morning, seeing nice, lush healthy plants. Id come home from work , later that day , to see wilted plants. Id go to check them, pulled lightly, and the whole plant would pull up (easily). I poked my finger into the soil where the plant pulled out, and sure enough, my finger entered the mole hole/ tunnel.
I eventually planned a new raised garden. This time I placed 1/2 inch X 1/2 inch wire mesh. It has worked %100.
As mentioned in an earlier reply, the regular chicken wire holes are too large. If what you have are moles or voles, they can easily fit through them. The 1/2 X 1/2 wire is usually a little thicker gauge wire too, which would make it much more difficult to chew through.
If you have ground hogs or gophers, im not sure if the wire will be much of a match for their teeth.
All this being said, when I made my new raised bed, a significant amount of construction was done, which included a lot of digging up/ leveling the land and bringing in truck loads of dirt. So, although the moles have disappeared, the demolition and bringing in the new soil may have eliminated the problem, even before the wire had a chance to work.
Also, if the wire used is not galvanized, it will rust easily, and probably only get a season or two out of it. Galvanized wire will last much longer, not sure of its affects on the soil.
Finally, make sure you have a significant amount of soil covering the wire. Problem #1, if you use a shovel, and kinda forget or are two aggressive, you will go right through the wire. And believe me, its a pain in the butt to have to remove all the soil just to get another layer of wire in. Problem #2, the roots will grow through the wire ( which is good), except when you try to pull the plants out at the end of the season or during harvesting. If the roots are tightly around the wire, could damage the wire or the root vegetable as you pick them.