I'm not sure I can explain this, and I'm not sure it would work with the design of the dresser, but you could build the calendar boxes around a hollow tube of some sort - even a drinking straw would probably work - and attach a knob to the top of the box/tube assembly. Then take the base of the box and attach a thin dowel/bamboo skewer that is smaller around than the straw. (I'm not guessing how to do this attaching - just that the skewer needs to be attached to the base somehow

) Then, if you set the box/tube assembly on the base so the skewer goes up thru the tube, you'd be able to use the knob at the top to turn it. If you rubbed some wax paper on the bottom of the box assembly and the top of the platform, it would be easier to turn.
This assembly won't be as substantial as attaching a turntable, but it also would be considerably less expensive. It all depends on what you are planning to use your calendar for, and how long you want it to last.