How to create a rotation animation with "Manipulate Centre Points" turned on?

Question

I'm trying to create an animation of an object rotating around a center point. By setting the 3D cursor a distance from the object, turning "Manipulate Centre Points" on and rotating freely, the rotation works just as a I want. For example:

The problem I'm having is when I try to animate this. When I rotate the object and try to set a "rotation" key frame, the rotation doesn't work. I think this is because "Manipulate Centre Points" does not actually change the rotation, only the location:

I've managed to create a work-around by rotating the object small amounts, then creating "location" keyframes for each step. But this an approximation of the rotation, because the object is actually moving in a straight line between keyframes rather than in a true circle. Is there a better/easier way to do this?

Edit: Omar Ahmad, Xtremity & PGmath's answers below all work perfectly. PGMath has some animated GIFs in the answer, for which I clicked "accept".

Answer 1

You can do this by parenting the object to an empty and using rotation constraints.

First add an empty at the point you want the object to rotate around. Then select the object and then the empty (in that order) and hit Ctrl+P to parent it to the empty. Now you can animate the empty revolving around as desired.

Then select the object and go to the Constraints Properties panel and add a Limit Rotation constraint and enable the constraint on all three axes, leaving the Min and Max values at 0.

Answer 2

You can use "Follow Path Constraint".

Add a circle curve,then select the object and add the Follow Path.

Press Animate Path and check follow curve if needed,and play the animation.

Answer 3

Add an Empty to the scene at the origin. Set Suzanne's parent to the Empty by selecting Suzanne, then the Empty then pressing Ctrl+P. Rotating the empty should now rotate Suzanne about the origin. However, the monkey turns to face the origin which isn't desired. Now add a Limit Rotation constraint to the Suzanne object. Enable Z-axis, plus any others you need. When you rotate the empty, Suzanne will keep facing south.