I've played Skyrim for 220 hours without once swinging a two-handed weapon. Since I found myself bored with my sneaky archer and feeling curious, I decided to start a new game, aiming to create a Paladin class (Two-Handed, Heavy Armor, Illusion, Restoration) with a Khajiit.
Making my way through the fort at Helgen, crushing skulls with bare hands, I encountered my first two-handed weapon: an iron greatsword. With bruised knuckles and a firm grip, I took up the rusted blade amidst battle, swung mightily at a villainous Stormcloak, and smote him down to the dirt from whence he came. With the twang of the blade still resonating loudly from the blow to his temple, I engaged another Stormcloak, this time with the bloodlust of a maniac.
Forget their slow, encumbering nature. Forget the vulnerability you feel. The gods gave you two hands. Use them both for your weapon.
I'm using mods which add more realistic two-handed swords (faster and less bulky), as well as the ability to run with it in one hand, and only use both hands when you actually attack, and it makes the gameplay great. There is the obvious downside of not being able to use spells while fighting, but the power of two handed weapons, and their ability to 'shield bash' more than makes up for it imo.