We construct p-adic L-functions associated to cuspidal Hilbert modular eigenforms of parallel weight two in certain dihedral or anticyclotomic extensions via the Jacquet-Langlands correspondence, generalizing works of Bertolini-Darmon, Vatsal and others. The construction given here is adelic, which allows us to deduce a precise interpolation formula from a Waldspurger-type theorem, as well as a formula for the dihedral mu-invariant. We also make a note of Howard's non-vanishing criterion for these p-adic L-functions, which can be used to reduce the associated Iwasawa main conjecture to a certain non-triviality criterion for families of p-adic L-functions.