We characterize the asymptotic behaviour of the weighted power variation processes associated with iterated Brownian motion. We prove weak convergence results in the sense of finite dimensional distributions, and show that the laws of the limiting objects can always be expressed in terms of three independent Brownian motions X, Y and B, as well as of the local times of Y. In particular, our results involve "weighted'' versions of Kesten and Spitzer's Brownian motion in random scenery. Our findings extend the theory of stochastic integration developed theory initiated by Khoshnevisan and Lewis (1999), and should be compared with the recent results by Nourdin, Nualart and Tudor (2007) and Swanson (2007), concerning the weighted power variations of self-similar Gaussian processes.