Most quizzes will accept cash but there is an increased risk in sending money that way so if you enter many quizzes, I would say keep one cheque account active. Many people have accounts with both a high-street bank and with a building society. By all means go green on one and reduce statements on the other to quarterly instead of monthly.
As for direct bank transfers, that is fine when done through large organisations with secure billing pages - account details are then kept private and safe. The reality is that most charity quizzes are run by individuals or small groups on behalf of a larger charity. These people are unlikely to have secure websites and would rightly be reluctant to disclose bank account details on an open web page. I would even be wary of including it in just the quiz itself, though having said that, when you send a cheque, it has your bank account number on the bottom and you don't generally think twice about that.
Finally, If payments were to be made by direct bank transfers, it would add an extra layer of checking before the quiz-setter could send out the quiz. When I was doing my quiz, most days the quizzes were sent within 2 hours of the post arriving. Each one simply required the recording of date, name, address and amount received. Having to access an on-line bank account and match up payments with requests would further delay this process.
Oh! Just realised why that wouldn't work. Payments are generally made direct to the charity involved, but the individual(s) running the quiz would not have access to view that account on-line. Therefore the idea is a non-starter. (I don't think people would be keen to pay money into someone's personal account when it was intended for a charity.