I find this time of year to be the most difficult to protect young seedlings. Everything, summer-like is still in full swing. It could be lizards, birds, snails, and/or pill bugs.

My past strategies have been mainly - cages, with wire around them (for skunks, birds mainly), pieces of screen (birds - especially over emerging snow pea seedlings, and smaller fry - possibly lizards). I've also tried hiding seedlings among the pepper plants or other larger ones.

But one option that seemed to work best - was to wait until cooler weather. The birds go south to the playas, the lizards stop being active, as do a lot of the bugs. That, of course, leaves snails, pill bugs and the white butterfly larvae.

Soooo - yes screens work OK. A hoop type pvc cage with Remay cloth works OK. Waiting to plant seedlings when they get bigger - disinterests pill bugs. Hand picking worms and snails - I'm afraid is all I know to do - organically.

But - like you, every once in awhile I take my chances and plant them anyway - OK, so I'm starting my second round of bok choy and leeks again. On the bright side, I have another batch of leeks seeking shelter underneath 2 nursery flats - to provide not only some shelter, but a bit of cool - cuz the temps are NOW in the 100's. (OK - so where were yu when the tomatoes needed ya).

Bejay

A bluejay kept taking my spinach. I saw him, so I know it was him. I had to put a cage around the spinach.

My quail, whom I encourage and nurture, will eat lettuce seedlings. I put some hardware cloth over lettuce when I plant it.

I am afraid to use bird netting for fear I'll capture the birds and injure them.

Once the seedlings are up past about 3 inches tall, the quail lose interest.

So, you could be losing your seedlings to birds. Crows and ravens make good suspects. They would have picked those plants up and flown off with them.

Some birds will pull the plant up to get the seed. They'll do that to corn and sweet peas.

Not much chance that this is your culprit, but I had an escaped iguana eat my grape crop this year.

I would have guessed rabbits under the assumption the plants were taken at night. But you said you don't have rabbits. So maybe an early morning squirrel?

I've had voles clean out every broccoli seedling--set out under row covers.

I think the most likely culprit is cutworms. They cut your plants off at soil level. Sometimes they leave the plant lying on the ground, but some cutworms eat the whole plant.
They're most prevalent if you've just turned over grass to make your garden.
You can use cutworm collars to protect your plants from cutworms.
Of course a rabbit might be sneaking into your yard at night.

Thanks everyone for your ideas...we found the culprit. We have a rat in our backyard! My husband is feaverishly trying to catch him. I believe we have a bionic rat. He has evaded all of our snap traps and glue traps. We do not want to resort to poison since we have animals.