Cash payment is direct, so in cash, not by cash. Mind you, some people I know would say 'by cash'.
All the best.
Check out this source: HLT Magazine (May 2001) - Seth Lindstromberg column
"...if you pay by check, the check is a means by which you pay. But, it is not payment in itself! Not yet. The person you give the check to receives payment only when the check is cashed. So the check is between the beginning and the end of the transaction.
A cash payment is different. The cash is the payment; there is nothing in between. (The same is true of so-called 'payment in kind'--which is when, for example, to buy 20 potatoes, you pay with 1 chicken.)
The term, payment in cash, then, means 'payment in the form of cash'. The cash is the end of the transaction. If you ask: "Where is the payment, where is the end of the transaction?" The answer is: "Look. It is here. In this cash."