Once you have the shape and location of your new garden all set, it is time to take care of the turf. Below are some simple methods of grass removal.
Dig & Flip (The Upside Down)
This method of grass removal is great for small areas. Grab a shovel and dig down 4-5"; then flip the soil so that the grass lands underneath. Continue to do this in a systematic way, creating rows until sod (and the roots of your sod) is no longer visible. Let the soil dry slightly and then tamp down by stepping on the area. Rake to smooth the loose soil and fill the gaps. It may take a bit more effort to dig through the grass layer underneath when planting, but the decomposing grass will add a bit of organic matter and moisture to the soil—this is good.
If you have the time (2 months or longer), save your cardboard and give this method a try. Flatten all saved boxes and layer them in your new garden area. Make sure that all seams are covered by overlapping your cardboard. If you have access to mulch (free mulch is available from City of Chicago), cover your cardboard with 2-3" to help hold moisture in and break down your cardboard. If you do not mulch, weight it all down with several heavy rocks or bricks, and remove your cardboard before planting. There may be some residual weeds like dandelion that you can pull or dig before planting.