Cats will try to dig up the voles! But it's true, they can be great vole-reducers. They're kind of hard on the local songbird population, though. Maybe you could borrow a neighbor's cat & have it in the garden when you are working in there!
Cats don't dig for the reasons dogs such as terriers do (for the heck of it)---they dig to bury their poops, and to try to get at an animal in a burrow. Cat poop is not fertilizer so you don't want them to make part of your garden a litter box, but a bit once in a while here & there isn't going to kill everything.
There are other vole deterrents--if you google around you'll find suggestions to use capsaicin spray (mammals can taste hot peppers; birds can't)and other noxious tasting & smelling things, as well as the mesh suggestion.
Traps might not be awfully successful---why mess with your suspicious looking trap, when there are far better looking things to eat all around the garden?

The thing about sprays is that they have to be re-applied, and people say the critters get used to them eventually. Anyway a plant I'd covered with pepper/wax spray was missing another few leaves today. I sprinkled with hot pepper but we're supposed to get 2-3 inches of rain in the next couple of days.

Good point about the songbirds. I could fence in an area surrounding the garden and have a cat patrol that! Not practical for right now, though. Anyway they might just tunnel right underneath!

The mesh seems like a good idea. But I think that a mesh fine enough to keep out voles (½") would also keep out the snakes, toads, etc. who keep down the slug population. No other solution for that unless I put a few geese or ducks in the garden!

Has anyone else successfully dealt with voles? (They are also known as field mice.)