You really don't need to request a rotation.
DCL assigns everyone to a rotation. And everyone onboard will have the same opportunity to dine in all 3 dining rooms, with the same menus during the duration of the cruise.
That is not entirely true on the 7 night Fantasy. Since you have the animation show the last night that you are in Animator's Palate (either Prince and Princess night, Captain's Gala night, or See You Real Soon night) you get a reduced version of the menu that night. Instead of a choice of appetizer you get a sampler trio (not sure if it varies based on the night), one soup (you get none of the usual choices for that night), and only 1 choice of dessert (or the Sweet Temptations). Luckily, you get the full entrée menu. So if there is a appetizer, soup/salad, or dessert that you don't want to miss, and it's not available at Animator's Palate, be sure to request a rotation that does not put you in Animator's Palate for that night.
In the case of my recent Fantasy Eastern cruise, my wife requested ERAERAE so that we could be in Royal Court for the first formal night (although it put us in Enchanted Garden for pirate night and Animator's Palate for the semi-formal night, neither of which make any sense). Initially, I wanted AERAERA since it would put us in Animator's Palate for pirate night, and Royal Court for the semi-formal night (captain's gala / lobster night), with Enchanted Garden on formal night as sort of a compromise. But when I looked at the menus, if we had been in Animator's Palate for See You Real Soon night, and would not have been able to order the chocolate decadence, since that is not the one dessert choice you get that night in Animator's Palate. However, if you are in Animator's Palate on captain's gala night, the one dessert choice is the warm chocolate lava cake, which is the dessert we both would have likely chosen that night anyway. So I went with my wife's original suggestion of ERAERAE since it seemed to offer the best of both worlds: first formal night in Royal Court, and it gave us access to both the warm chocolate lava cake and the chocolate decadence.
It's been speculated that by not requesting a specific rotation, you have a better chance of getting table mates that match your group demographic. I don't know if that's true or not, but, we've never requested a specific rotation, and always had excellent matches in our tablemates.
I had initially not wanted to request a rotation for that reason. Interestingly, we had a table for just the 2 of us. I wonder if that was because we were an adult couple with no children, but requested a rotation that they probably assign children to (and maybe a lot of families with girls who wanted to be in Royal Court for Prince and Princess night?)
But, yes, you can request a specific rotation. DCL will point out that it's just a request, but it can be added to your reservation. Most times such requests are granted. Sometimes not.
We have requested a rotation exactly twice - to skip the Crush show in AP and do Palo instead - and both times we were sat alone instead of with table mates. We didn't request to seat alone. It was actually really odd when we were expecting to have table mates.
Like I said, I wonder if that was why we had a table for ourselves, since we requested a rotation that we otherwise would not have been assigned to.
Last year, on the Dream, my wife wanted to request a rotation but I didn't, since I felt Disney would be better than us in assigning rotations. When we got ERRA, my wife was excited, thinking we were in Royal Palace on formal night. But then it turned out the 2nd night was Halloween night and the formal night was the 4th night. Though I don't know whether or not she would have gotten what she wanted, since she would have likely asked for ERRA.
We have also had great matches in our table mates and cruise once a year with a couple we met in 2013 on Disney.

It's been awesome.