First, all the newest ships have all the latest computerized stabilizing devices. Because of this, they are now alot more stable than cruise ships of old.
Now, with that said, it does not mean you won't get seasick, but chances are fairly slim. And any movement that you get will be completely different that the movement you get on smaller boats. I know people who never get seasick on a small boat but will on a big ship. And I know people who get seasick on a big ship but never on a small boat. Obviously, it affects everyone different. I spent 6 years in the Navy and the first time we ever hit a bad storm and I was on a small ship, I got seasick for 3 days! I would not have been surprised to see my shoes coming out of my mouth I was throwing up so hard!! But as they say, once you get seasick that bad, you'll never get seasick again. And they weren't kidding - I've never gotten seasick or motion sickness again.
They did an episode on Mythbusters about seasickness that I thought was very informative. They tried several different remedies to see which one worked best. They tried the wristbands, Dimenhydrinate (Dramamine is one brand), non-drowsy Dimenhydrinate, Scopolamine patches, and ginger tables. What was most interesting is that they had the best results with simple ginger tablets, which the Chinese have used for thousands of years to help prevent motion sickness. You can get it cheaply over the counter at any drugstore. Who knew?
The thing about any of these medications is that you have to begin using them about 3 days in advance. If you wait until you need them, it's too late.
Alot of people have adverse reactions to the various drug items, which can ruin your vacation, so personally I would want to try the non-drug items first since there are no side affects with them.
If you have a problem with motion sickness on small boats does not necessarily mean you'll have any problems on large cruise ships or vice versa. We were on the Allure of the Seas and at times it was really hard realizing we were even at sea. It is an extremely stable platform, that's for sure.