We present integral field spectroscopic observations with the Potsdam Multi-Aperture Spectrophotometer of all 14 galaxies in the z similar to 0.1 Lyman Alpha Reference Sample (LARS). We produce 2D line-of-sight velocity maps and velocity dispersion maps from the Balmer alpha (H alpha) emission in our data cubes. These maps trace the spectral and spatial properties of the LARS galaxies' intrinsic Ly alpha radiation field. We show our kinematic maps that are spatially registered onto the Hubble Space Telescope H alpha and Lyman alpha (Ly alpha) images. We can conjecture a causal connection between spatially resolved H alpha kinematics and Ly alpha photometry for individual galaxies, however, no general trend can be established for the whole sample. Furthermore, we compute the intrinsic velocity dispersion sigma(0), the shearing velocity v(shear), and the v(shear)/sigma(0) ratio from our kinematic maps. In general LARS galaxies are characterised by high intrinsic velocity dispersions (54 km s(-1) median) and low shearing velocities (65 km s(-1) median). The v(shear/sigma 0) values range from 0.5 to 3.2 with an average of 1.5. It is noteworthy that five galaxies of the sample are dispersion-dominated systems with v(shear)/sigma(0) < 1, and are thus kinematically similar to turbulent star-forming galaxies seen at high redshift. When linking our kinematical statistics to the global LARS Ly alpha properties, we find that dispersion-dominated systems show higher Ly alpha equivalent widths and higher Ly alpha escape fractions than systems with v(shear)/sigma(0) > 1. Our result indicates that turbulence in actively star-forming systems is causally connected to interstellar medium conditions that favour an escape of Ly alpha radiation.