... a check with my routing number and account number in the "Pay to Order of" field
I understand your desire for anonymity, but this is not a good solution. You are exposing your account number and routing number to anyone who might see the check.
Having them make the check payable to "CASH" is certainly a good solution, but it requires that the person writing the check "trusts" you highly. The problem with Writing a check payable to "CASH", is there is no guarantee that the person that eventually cashes the check is the same person that the check was written for.
Consider these 2 scenarios:
Person 1 (John) writes a check made payable to CASH, and gives it to Person 2 (Jane) as payment for an item or a service. At some point, Jane loses the check and someone else (Bob) finds the lost check and cashes it. John would have no recourse (or very little) against Bob for cashing the check, and Jane could claim she was still owed the money from John. John's check was paid, but he has nothing to prove that he paid Jane.
John writes a check made payable to CASH, and gives it to Jane as above. Jane is really dishonest and she gives the check to her friend (Sally) who lives in another state. Sally deposits the check and Jane claims she never received payment from John.
When you deposit the check (or cash it), you will have to (endorse) sign your name on the back of the check, and the check (or an image of it) will be returned to the person who wrote the check. For many (perhaps most) US banks, endorsing the check might be optional, but probably will be required for a check made payable to CASH, or to "routing number and account number", so using either of these for total anonymity probably won't work.
The other option would be for the person to pay you using "Bill Pay" from their checking account, making the payment to "You" (your name) with your checking account number and bank name and address (and possible routing number). This payment will be made electronically through the ACH system and there will be no paper check or copies of it returned to the person making the payment. And the deposit will be made to your account without your signature.
Keep in mind that a copy of a check when it was written or after it was cashed, is not the only place your name might appear. Even if you are paid in actual cash (currency) there is likely to be other records existing like receipts or purchase orders that will have your name on them.