Complete the main storyline, then complete the following task to unlock the corresponding bonus dungeon.

Bufu Cave
Beat Main Story (Talk to the Tonjiro in the Elder's House and select "I have no idea.")

Dragon Veins
Beat New Chapter (Examine Yotsume Rock)

Hidden Path
Complete Master Rank of the Battle Arena

Millennium Dungeon (Karakuri Inn)
Beat New Chapter (Examine Yotsume Rock)

Mysterious Dungeon
Beat Main Story (Talk to the WIS Staff in the Tavern)

Ochimizu Pass (Extend)
Beat Main Story (Examine Well)

Sanuki Thicket (Extend)
Beat Main Story (Examine Well)

Shrine of Mystery
Beat Main Story (Talk to the WIS Staff in the Tavern)

Void of the Strong
Beat New Chapter (Examine the Blacksmiths Stove)