The blackbirds will be after the worms, so as long as you don't kill THEM off I don't think cutting the grass is going to harm the blackbirds.
We also have dogs, and perform the "pick up" ritual before each mowing, which involves me going round with a load of plantpots, marking the dropping zones, and my OH or his son following behind with the dustpan and paint scraper. The contents are disposed of down the loo.
Our grass is a mixture of many 'weeds' including daisies, ajuga reptans etc. We try to maintain a balance so that the large-leaved weeds do not take over. Other than that, Nature does pretty much what she wants, and we just give her the occasional haircut.