Sow your seed BEFORE you chop and drop the existing vegetation.
If it's seed that requires light to germinate, remove the mulch until your crop is ready for it.
One alternative that works well in a climate with fairly soggy springs is scorching by weed-torch. These can be fueled just as easily by home-distilled ethanol as by fossil fuels. Note that this is scorching, NOT burning. The point is to apply
enough heat to kill the plants by internal boiling, not to chemically alter their structures and send part of them up in smoke.
EDIT: just a reminder if you chose the weed-torch route, be careful to focus the heat into the higher parts of the plants [most winter covercrops get quite tall, thankfully] so you aren't damaging the soil life too much.