I attached a couple of photos of the holes. They are sometimes many in a row less than 12 inches of space between. 

You took the words right out of my mouth, “holey moley”. There is certainly something invading your garden!

You are right, this is not likely to be moles. Moles leave a mound that is round and forced up from the center. Their mounds never have open holes. The holes in your pictures are definitely open and do not have mounds around them.

Pocket gophers can also be ruled because they also leave mounds. Their mounds are fan shaped because the dirt is thrown out in one direction. Their holes can either be open or plugged with the closure off to one side.

The most likely culprit are voles. Voles, or meadow mice, grow to about 5 inches long and have brown-black fur. They spend most of their time underground or in dense grass. In areas of heavy activity, they leave open 1-inch holes, with no mounds, and form trails between the holes. They are active year-round and like to eat grass, as seen in your yard. They will eat lots of other things, including small grains, vegetables, and tree bark.

There are several ways to help control voles in your garden. Keep all grasses mowed and short, this includes around any trees you may have in your yard. Prevent or eliminate thatch that voles like to hide under. If you have root vegetables in the ground this winter or fruit that might have remained on the ground, remove them. Voles are encouraged by a nice bed of loose mulch, so be careful if you use it, particularly around trees and shrubs. Also, all rodents love bird feeder seed. If you have a bird feeder, keep as much seed from falling and remaining on the ground.

One last way to control voles is by using mousetraps. Bait them with peanut butter or apple and dig into their tunnels. Place the traps and cover them with boards. This can help to control small populations of voles.

For more information on voles: Moles, Voles and Gophers Dig the Garden.