I looked closer and discovered many tiny black little beetle-like bugs. They appeared to fly away if I bothered them. They are flea beetles.
I have since learned that they are actually jumping away (like fleas). Plants that attract these beetles are cole crops, corn, eggplant, grapes, potatoes, spinach, sweet potatoes, tobacco, tomatoes and cucumbers. The beetles' larvae overwinter in the soil around their preferred crops.
Oregon State University Extension offered a lot of chemical control options (not an option for my organic homestead) in addition to the suggestion of planting "trap crops" of radish or daikon. The idea being to distract the little beasties with a plant they like better.
My infestation of flea beetles seems to like potato, tomato and pole beans. I have been treating the leaves of my plants with a combination of castille soap, neem oil and water. It seems to work to keep the little buggers running, but they come back even though I have been spraying a couple times a day. Other websites suggest to dust with diatomaceous earth (fossilized remains of diatoms, hard shelled algae). DE is extremely abrasive and breaks down insect exoskeletons. I have not used this yet, as I thought it better to not hurt any potential beneficial insects, but I may try it once before the toms start flowering. (I keep DE on hand for controlling fleas/mites in my chickens bedding.)
But first, I am going to try to find some radish or daikon starts - if there are any to be had at this time of year. We are not big radish fans in our family, so do not really care if the little fleas eat them!
Another tip I stumbled across was the use of mint or catnip, which seems to repel them. Luckily, I have a lot of mint:
So I harvested some stalks and placed them alongside the tomato plants. I will let you know how it goes.