Last year we planted Rye and winter peas as cover. Probably won’t plant a “grass” as cover again-roots are hard to break up. Also, we could have turned it over earlier, I suppose, but grasses have more extensive roots. But the peas were good and don’t csuse any problems.I still won't plant a cover crop again.
In another bed we used yellow clover as cover. It worked well - spread seed in spring, tilled under by June/July. But this was in the big bed that can be tilled with big tiller. We still get small clover plants at the edge, but we pick them for the chickens, so they don’t go to seed.