The simplest option in replacing Trapsense would just be to substitutue it with a Bonus Feat from the Fighter list, except maybe only give four over the 20 level progression. That gives them greater versatility but doesn't really infringe on the advantages of a single class Fighter.
Another option would be a feat/ability tree, similar to Rangers and their combat style. I would suggest limiting it to four feats/abilities over the 20 level progression. Space them out like 3rd, 7th, 11th, 16th or whatever you want (I don't have the PHB handy so I can't tell off-hand when they would normally get the Trapsense ability).
Something like:
--Quick Barbarian: Dodge, Mobility, Lightning Reflexes, Evasion.
--Tough Barbarian: Toughness. Dwarven Toughness. Giant Toughness. Dragon Toughness.
--Fast Barbarian: Woodland Stride, Increase move +10', Run, Powerful Stride (Just made up the name--move not reduced by medium/heavy armor or encumbrance; just like the dwarf racial ability).
--Hunter Barbarian: Nature Sense, Track, Swift and Silent (the feat, but limited to natural surroundings), Swift Tracker.
--Furious Barbarian: Various Rage feats (Destructive Rage, Instantaneous Rage, Extra Rage, etc.).
That should provide enough versatility for dungeon and non-dungeon going Barbarians can enjoy.