[dice]

I chopped mine down when I saw the first grain head. I raked
off the top growth to save it for mulch (it was too early to just
leave the mulch in place, the soil needed to warm up first).
In some beds I turned the stubble over with a spade (needed
more organic matter down deeper in the bed, and I had spread
alfalfa, shredded leaves, gypsum, and dolomite lime over it).
In others I only turned it over right where a planting hole was
going to be, leaving the spaces in between the holes unturned
with the mowed rye stubble sticking up.

It seems to work fine both ways. I am getting a few sprigs
still growing, probably late plants that had not flowered yet
when I mowed it. Easy enough to pull them out by hand or
hoe them out when the soil is dry and leave the rye plants
lieing on top of the soil, where they will dry out in the sun
and die.

I think you can plant into the bed where the rye was growing
any time after it is mowed. It is going to take a couple of
weeks for the seedlings to grow roots out into it anyway.
I would turn it over anyplace that you are direct seeding and
plant into the dirt that was underneath the stubble (rye
suppresses sprouting of some kinds of small-seeded weeds,
and that probably goes for things like lettuce, carrots, arugula,
etc, too). If you are planting seedlings, just making a hole for
the seedling seems to work fine (especially if it is an "Earl's
Hole", with various soil amendments mixed in under and
around the seedling).