A flat footed character loses their dexterity bonus to their AC. Is a Barbarian's Unarmored Defense still considered if caught flat footed? Perhaps just the Con bonus? Help?
There is no concept of "flat-footed" in 5th Edition.
Most situations that would have given flat-footed (blinded, paralyzed, petrified, prone, restrained, stunned, and unconscious) in 3.5 instead give the attacker advantage on the attack roll.
Flanking and facing are available as variant rules in the Dungeon Master's Guide, and they, too, grant advantage on attack rolls.
Advantage is a simpler way to grant the bonus; it's fewer stats for each player to keep track off. It does reduce the difference between the various armor types, and if your table wants to houserule flat-foot AC into 5e, I would recommend that you remove the advantage from those status effects to compensate.