When I moved to a new place, I couldn't pay for anything with a check because I still had my old address printed on the checks and most stores wouldn't accept checks with an out-of-state address printed on it. This was ~18 years ago, so things may be different.
What systems to handle it?
The cashier puts the check under the cash tray like you do for $100 bills, and then you toss the check in the cash bag that you send to the bank that night or at the end of the week or whatever, and the bank handles the check.
I guess I was doing it to service providers rather than stores though - most stores would much rather have had a card payment even back then.