I am a tenant who pays cash into my landlord's account.
I do this for two reasons:
1 - I prefer to make sure the money has gone in...I once had a very angry landlord telephone me to accuse me of not paying my rent in one month (had been set up by standing order) I had to faff around ringing banks and eventually go into town and speak to someone over the counter, it turned out the account the landlord had requested to be paid into was an ISA and he owed tax from the previous year for going over the allowed tax-free amount. The money which had been taken from my account was being held in the bank's holding account. Now I like the idea of physically handing the cash over and getting a receipt.
2 - I am one of those people who prefers to take out cash and pay all my bills this way so I know exactly what's going out of my account and when.
I have to say though that I would never refuse to set up a standing order to pay rent. If my landlord prefered that method then I would do just that.