You could have wire worms, soil dwelling larvae of an above-ground beetle. They do much damage to many types of both above- and below-ground crops. I have the most trouble with them with my beans and carrots.
I get wire worms off and on in my raised beds. They're about 1" long, about 3/16" wide, a very glossy 'tea' color and very obviously segmented. The have dark colored mouthparts at the "front end" and three sets of dark short legs . I find them while distrubing the soil during cultivating or transplanting.
As I garden without synthetic insecticides, I heat up my raised beds (which haven't gotten planted yet this spring) by covering them with thick clear plastic and letting the sun's rays raise the soil temps to a range over several weeks to kill the larvae. I have 6 raised beds, and during the fall/winter, I don't use them all. I try to plan it out so I can solarize two or three beds on a rotating basis each off-season. It does seem to help. I also cultivate the soil often with a hand tool during the winter to expose the soil and let the birds find what insects they can.
Hope this helps. I also try to grow herbs and flowers within my raised beds to encourage predator insects and birds ( which means sometimes I have to drape my crop plants with row cover or cheesecloth lest they become too enticing to the birds)
For cutworms, you could try inserting a bamboo skewer (kabob stick thing-a-ma-jig) into the soil near the base of the plant. And, if it is something larger is doing the damage, like a critter, you could try planting them in little wire cages, as if you were rabbit-proofing them. Also, if you are not squeamish about slaughtering slugs, you could apply a little sluggo around the base of the plants. It is non-toxic which means it also washes away with overhead watering or rain. They say it is safe for wildlife and pets and so I use it for my strawberries.
We have 3 different kinds of slugs here that like our beets and turnips and radishes as well: the typical brown slug, some kind of almost bright orange slug running about the same size as the typical brown one, and a small brownish-black slug that's very difficult to see and we only happened to spot with luck. The brown and orange ones turn up regularly in our beer ponds but we've yet to see the little camouflaged ones in there. We've recently surrounded stuff with diatomaceous earth and that may be helping. We'll see when we pull our next root crops. Are the tops being attacked as well as the root? Is the damage you're seeing little round chomps nibbled out on the roots mostly around the top of the veggie? It might also be cabbage worms as well. We're still fighting them in our plastic covered raised beds. We still have broccoli growing which they really like but I think as we harvest the broc plants they're moving on to the next dish on the buffet.