If you plan to leave the manure crop on through the summer, mow or cut it each time it begins to flower, letting the clippings fall to the ground. That way, it can't go to seed and create a weed problem. When you turn the manure under the soil depends on the results you want. Grasses and other non-legumes add mostly nutrients if you turn them under when they are young and tender. When they are older, they become a better source of organic matter.
Mature legumes add more nitrogen than young ones, but can be harder to cut and turn under the soil.
A few weeks before you plan to plant your garden, use a scythe or other tool to chop the leaves and stems, or mow with a mulching mower. It will be easier to work the manure into the soil if you first let it dry for a few days. If you covered only a small area with a green manure, you can turn it under with a spade. For a larger area, use a rototiller. Tough plants such as old alfalfa may require a few passes.