Fast Travel isn't teleportation
Oblivion's very much is. Sure, time goes forward after you reach the destination (not before or during), but
nothing in the world affects you between the start and end of the "journey". The character does not exist in between the two points. It really isn't very different from teleporting then hitting the wait button. If you contend otherwise, then how is it different?
That said, I don't particularly care for the way Morrowind did it, either. It was better presented, but the mechanics were still the same underneath even though you had to talk to specific people to activate it. I actually think the way Oblivion did it was closer to the "ideal", but it needs some changes:
1) Risk. Random encounters that you can't avoid. Not all attempts at fast travel need to result in a random encounter, but the likelihood of an encounter should depend on how dangerous a given area is (deep forests being more likely for encounters than main roads, for example). Your stealth and visibility (light/chameleon effects) can also play a role. When you get such an encounter, the enemy automatically enters combat.
2) Proper simulation. Your character doesn't fly in a straight line in an instant. You could have the option of traveling on roads (safer, longer travel time), or through wilderness (dangerous, shorter travel time), but a proper path must be simulated. No auto-healing, and spell/potion effects that run out must happen at the proper point in the journey, not at the end (eg, no using a 1 second invisibility to be invisible for the entire trip).
3) Interruption. You're not a ghost. Any persistent enemies, or NPCs trying to interact with you, that you meet on your journey should automatically interrupt your travel as appropriate. If vampirism is in, then you should be interrupted if traveling when daybreak hits and you start taking sun damage. If lycanthropy is in, then your travel is interrupted on the appropriate nights when you change.
Ultimately, fast travel should not cause drastically different results than if you had travelled the same route manually.