You tip room service with cash. The room service food, coffee, and tea, are included in your cruise fare and have no extra charge. Hence you are tipping for the delivery. The exception is for midnight to 5am where they will have the $3.95 late night service charge. Since that includes the tip, all you need to do is sign it. Feel free to throw in a buck or two if you wish. It won't hurt their feelings.
Whoever told you there is no cash tipping is not really correct. You can't use cash to buy things other than in the casino, but it is the common form of tipping on Royal Caribbean - mostly on the last night of the cruise. I usually drop some cash at a couple of favorite bars on the first day too, since it helps me get remembered for the rest of the cruise.
Bar receipts also have a blank for an added tip, so you can add some there which will be charged to your account. I prefer to just hand a couple favorite bar tenders or servers some real money.
Generally speaking, for a typical passenger you would only "need to" tip in cash for the above room service scenario. This assumes you just stick with the 15% add-on for bar sales (which goes on your card), and you prepay or have the main tips added to your account once aboard the ship, since both of these result in the vouchers being delivered. The crew then redeem the vouchers for their money. However most of us put cash in people's hands during our cruises. Not every day, and not each time we get waited on, but at least a few times.
It might help to know how these things work: As I said earlier, the "main" tips are given to your cabin steward and your dining room staff. These are what typically add up to around $70 per person for a 7 night cruise. The 15% for all bar orders is pooled and shared by all the bar personnel. Most do this with the cash tips people lay on them too. Similarly, the room service people pool all tips and share them.
Tips are the primary source of income for these people. Cruise lines pay little or no salary to service staff employees since they provide room and board for them.