The compost heap is its best destination at present, preferably mixed with 40-50% paper and cardboard.
Using mown grass as a mulch is a tempting way to smother weed seedlings but it encourages slugs and can rot into a slimy layer on top of the soil. Later in the season it may also contain lots of grass and weed seeds.
As long as you can add enough brown materials to the compost heap, at the same time as all that grass, it should transform into something more valuable.