Don't bother taking up the grass. Just put down cardboard on top of the grass and put down choice of mulch (see below).
Once the grass is dead and cardboard has broken down, scrape up all the good composted sod top soil into the raised beds. This will lower the grade and tend to pool water so put down woody yard trimmings and tough weeds in the paths as mulch, then cover with decorative mulch if you want (I don't bother

). If any weed grows, hoe or pull and just drop on the path as mulch to break down.
This year, I didn't want to do the drowned weeds bucket, so instead, I put any questionable weeds with tendency to grow back or re-seed in the middle of the paths where I would definitely trample them into the mud. They all break down and become available as more composted topsoil to scrape up into the beds. I'm tossing corn leaves and tassles in the paths as they dry up, and corn stalks too cutting into chunks with hand pruners since these are coming out one at a time from my small patch of popcorn. Later on, I plan to smash the big ones with a hand sledge.