(abridged) Angular momentum transport and accretion in protoplanetary discs are generally believed to be driven by MHD turbulence via the magneto-rotational instability (MRI). The dynamics of solid bodies embedded in such discs (dust grains, boulders, planetesimals and planets) may be strongly affected by the turbulence, such that the formation pathways for planetary systems are determined in part by the strength and spatial distribution of the turbulent flow. We examine the dynamics of planetesimals, with radii between 1m \^a 10 km, embedded in turbulent protoplanetary discs, using three dimensional MHD simulations. The planetesimals experience gas drag and stochastic gravitational forces due to the turbulent disc. We use, and compare the results from, local shearing box simulations and global models in this study. The main aims of this work are to examine: the growth, and possible saturation, of the velocity dispersion of embedded planetesimals as a function of their size and disc parameters; the rate of radial migration and diffusion of planetesimals; the conditions under which the results from shearing box and global simulations agree. We find good agreement between local and global simulations when shearing boxes of dimension 4H x 16H x 2H are used (H being the local scale height). The magnitude of the density fluctuations obtained is sensitive to the box size, due to the excitation and propagation of spiral density waves. This affects the stochastic forcing experienced by planetesimals. [...] Our models show that fully developed MHD turbulence in protoplanetary discs would have a destructive effect on embedded planetesimals. Relatively low levels of turbulence are required for traditional models of planetesimal accretion to operate, this being consistent with the existence of a dead zone in protoplanetary discs.