We study the existence and stability of multibreathers in Klein-Gordon chains with interactions that are not restricted to nearest neighbors. We provide a general framework where such long range effects can be taken into consideration for arbitrarily varying (as a function of the node distance) linear couplings between arbitrary sets of neighbors in the chain. By examining special case examples such as three-site breathers with next-nearest-neighbors, we find crucial modifications to the nearest-neighbor picture of one-dimensional oscillators being excited either in- or anti-phase. Configurations with nontrivial phase profiles emerge from or collide with the ones with standard phase difference profiles, through supercritical or subcritical bifurcations respectively. Similar bifurcations emerge when examining four-site breathers with either next-nearest-neighbor or even interactions with the three-nearest one-dimensional neighbors. The latter setting can be thought of as a prototype for the two-dimensional building block, namely a square of lattice nodes, which is also examined. Our analytical predictions are found to be in very good agreement with numerical results.