In 1D, we get translations. In 2D, we get rotations. Why doesn't a 3rd spatial dimension introduce anything.
Followup: Do higher dimensions introduce anything?
Are there rigid motions of interest which cannot be described as a combination of translations and rotations?
It's not quite clear what you are asking about, but I suppose you mean rigid motions (which preserve the distance between any two points), which also preserve orientation (which rules out mirror reflections).
If so, then as Ennar points out, there is a new kind of motion in three dimensions: the screw translation. This is a rotation about an axis combined with a translation along the same axis.
The simple rotation and the simple translation are both special cases of the screw translation, with the translation part or the rotation part being zero, respectively.