When I was first establishing my garden I has truck loads of compost delivered that I freely dumped on top of my beds. The past sevral years I've been making compost myself and can make about three yards, which is just not enough for my garden. I wanted to not be dependant on a truck to bring me compost every year and I've found that the rye adds a substantial amount of organic matter.
A good planting of winter rye, a wheel barrow of compost and a balanced mix of rock dust keeps the hard working soil in each bed healthy and productive.
Here is how I deal with the rye:
I start by loosening the soil with a broad fork. Then I start at one end and just yank it out by the hand full, and lay it down on the bed.
Here it is after 2 weeks. It's got a nice soft, crumbly texture. Ideally I will now add my rock dust and compost and wait one more week before planting. But I have been known not to wait that extra week. You can still see the little wisps of rye. They won't interfere with anything that's growing, and the rye will continue to decompose, adding nutrients and improving the texture of the soil.
Some years I wait too long, and miss my window of opportunity. If I find myself with knee high rye and tomatoes ready to go into the ground, I'll loosen the soil and pull the rye up and throw it in my wheel barrow. I'll add the rye to my cooking compost pile and add an extra load of finished compost to that bed. In the end it seems to even itself out.