If you don't have any up at all then it's going to be very difficult to move on any normal surface, that's just the physics of friction. OTOH, there are ways of moving with an up component that don't have the the body actually moving upward and bobbing up and down.
This should be a no-brainer for anyone who trains outside on a regular basis.
More to the topic - heel or toe or bubbling well - what mostly matters is the weighting, IMO. Also, I'd say that turning badly accounts for most (all?) of the knee problems that are so common among Aikido folks.
It took my internalizing and consulting with my instructors to understand the knee pain I had was the result of leaving my foot in place while the knee started the turn. The knee can't be ahead of the foot (torquing the knee), the knee and the foot turn as one. This is my current understanding... and my knees don't hurt anymore!
As far as up and down, I've been taught to turn without raising the hips (still working on it - 1st kyu here). You give something back to uke you (should) have taken earlier. I suppose that doesn't preclude the foot itself moving up and down with the knee absorbing the up and down to keep the hips from bobbing.
To Graham, as to my other comment on "forward" I meant the intent of the turn shouldn't ever be backwards. Same with pivots.
Working through this myself... not preaching