Here's the deal folks - if you do not want grocers, or anyone else, to know who you are and be able to track your purchases - pay cash for everything. No checks. No credit cards. No ATM/Debit cards. Nothing. Zip. Nadda.
The nifty IBM 4690 series POS computers (cash registers) and their competitors, have the ability to retain all purchase activity from a store - including items purchased, date, time, lane, cashier that rang the transaction, any and all discounts that were applied, coupons - and tender method. If the tender type is check, credit card or debit card, that account information is electronically stored (although not all retailers save the check's account information, they can, its a simple user exit on the controller). Paranoid yet? Good. How do I know this? Its what I do for a living - drive computers for a large grocery retailer/wholesaler that also provides technical computer support for independent retailers.
I sent a guy to prison a couple of years ago by blasting his alibi to pieces. He was charged with armed robbery, said he couldn't have done it because he was with his girlfriend except when he went to a particular store and got X, Y and Z and "some other stuff". Too bad for him we retained that information for 2 years. Spun through all transactions where X, Y OR Z were sold on that day, within 5 hours either way of the crime. Then looked for comparisons (eg., Oops, I got X and Y but forgot Z....) Bad news. NO Z items were sold that day AT ALL. No X or Y items were sold within the same transaction, and no transactions occured where either one or the other were sold within 2 hours of each other. When the defendent's attorney found out that we could pull that information back, and he didn't have a defense, never mind the fact that the accused was not seen on the security camera tapes for that day at all... Don't like armed robbers....
If you Yanks don't want Microsoft or AOL or whatever ISP you use to know who you are (as opposed to who you claim to be) and where you go on the net, do NOT give them (or the store where you purchased your computer in some cases) your Social Security Number. FOr a while that was all the rage - still might be for all I know. "We need your SSN to to credit checks." Not True. Also, not legal. Driver's license, State ID, Home Phone number, credit card number... any of those gives any credit researcher worth its salt the information needed to make a 90% id of you. Any TWO of those and you've got 98%+. Odds are, they asked for (and got) all three.
If you want the discount, give them a bogus name and address, get the card and get your discount. If you want to get coupons or save points, give them your real address. Don't want their computers to link your bogus name with your real name? Pay cash every time you use the card - ALWAYS.