You can put the straw in the compost and that's obviously easiest, rake it up and throw it in compost. But particularly if the straw seems like it is partly broken down/ softened up from the Minnesota winter, you can just turn it under, bury it in the garden that it has been mulching. What you would worry about is that straw might use up some of the nitrogen from your soil in the process of being broken down. But one thing I have discovered. I use fall leaves to mulch my gardens for the winter. In spring, I turn them under. Having a stash of fall leaves under the soil seems to be a big inducement to earth worms and I have way more of them now than I used to. So there's pluses and minuses. You can deal with the Nitrogen problem by burying something green with your straw, like grass clippings.