"to toast those seeds."
I don't know if you would toast those seeds under black plastic raised up that far above those seeds. If you mean kill the rye, again the plastic could only keep light out...a slow process perhaps for dormant plants. I guess I would till the stuff up daily until dead.

From your description, it doesn't sound like the cover crop self-seeded. Am I missing something? I'm not a fan of Roundup, but it might be your best bet in this situation. Wait until everything's dead and then til. You should be good to go.

Thanks guys. I sneaked a peak under the plastic and the cover crop, believe it or not, has sprouted through the compost in places. It is rainy now (so can't till) and I'm not sure if Roundup will work until the sun comes out warm for a few days; heaven knows when that will be. Should I leave the plastic down or remove it? That is leaving it down longer won't hurt the soil-food-web will it?

At least I know the ground is fertile!

My vote would be to leave the plastic down and plant through it if need be. If things are sprouting, then you have a viable seed population. If they sprout, but don't get light, they'll die and add to the nutrient base. In a few months you should be able to remove the plastic with no fear of further sprouting, and then finish your landscaping. If you had not already gone to the trouble of buying and laying the plastic, I would have planted my landscaping materials as the season warranted and done a weekly cultivating regime to make sure everything was killed back. Clear plastic is a better choice for heat killing seedlings, but it is probably still too early in the season to rely on enough sun for that treatment to be effective.

I have not found it necessary to cover any of my cover/green manure crops with plastic to "kill" them and since I did not want to "kill" them I would not have done it anyway. Mowing the crop, and then tilling it in has been enough. No plastic. No poisons. No anything that might possibly be detrimental to the Soil Food Web that would work on that cover/green manure crop.

kimmsr is right & has done this a lot longer then I.
I would not use round-up just to kill a annual rye grass.
Just till it under.
Mulch the bed if & when the rye comes up & it will die & rot into compost.