Assuming you have a sunny lawn, then water first, cover with plastic, and hold down the edges to keep the moisture trapped inside. You'll see beads of moisture forming on the inside of the plastic. Those beads magnify the sun and cook the soil and plants. It also cooks the seeds and, sadly, the soil microbes. When you pull the plastic off you might want to scatter some excellent smelling compost over to reestablish the microbe species. Then feed with organic fertilizer to reestablish the population of microbes from the compost. You don't need any more than 1 cubic yard of compost per 1,000 square feet. That would be a lot.

That's great but will it get all the deep roots or will they just regrow again. That is my only real question here.

Good question. I've not done it to verify. This is called solarizing. You might Google it. I realize that's why we're here (to know those answers), but organically killing a lawn is infrequently asked about. Most people use RoundUp, twice at 1-week intervals, and move on. If it were me I'd try solarizing for a couple of sunny days. Then pull off the plastic, water the lawn 3x per day very lightly for a week, and see if anything else grows. If more plants grow, put the plastic back down.

To kill seeds, temperatures have to be sustained over 106 for extremely long periods. Or 120 for somewhat shorter periods. Or 140 for a few days. Or 160 for a few hours. For those and for surface plants, it's a question of temperature and how long you have to keep up that given temperature to destroy the plant.

Root systems take longer. Days at high temperature, certainly. Plants with deep tap roots, like dandelion or butterfly weed or the like, will take much longer time periods.

Most people who solarize do so during the hottest part of summer and do it for weeks at a time. Temperatures under the plastic will vary from near air temperature just before dawn to extremely hot during a sunny afternoon.

...and you can cook a turkey at 120 degrees, but it takes hours...just to put it into perspective. The soil is a huge heat sink so it can dissipate the heat far and wide.

Thanks for all the information. You pretty much confirmed what I already suspected, although in much more technical detail. Solarization probably wont work for me. Our summers only get hot enough, like above 90 degrees, for a few weeks/year and there is no way to know if or when that would happen. Right now it's raining so much I think I saw someone building an arc. Since I won't spray chemicals, I think my only option is to rent a rototiller and dig it all up as best as I can.

You do have one other option which is organic/chemical. Horticultural vinegar, applied at the start of a hot, sunny afternoon, will take out most plants.

Be careful with the stuff, though, and wear enclosed eye protection like rear-venting goggles. Spray is acidic enough to severely damage your cornea, and that takes ages to heal (if it does).

That's something to consider. I was reading about vinegar recently as an herbicide. Again,my concern would be whether it will kill plants that have deep taproots. Will it do that, too?

Sometimes yes, sometimes no. Off the cuff, I'd say it does so more often than not if used appropriately (and the correct vinegar is used).

Household vinegar isn't strong enough, the stuff you want is horticultural grade.