According to a recent modified model of gravity at large distances, a radial constant and uniform extra-acceleration of Rindler type acts upon a test particle p in the static field of a central mass M if certain conditions are satisfied. Among other things, it was proposed as a potentially viable explanation of a part of the Pioneer anomaly. We study the impact that an anomalous Rindler-type term as large as m s−2 may have on the the orbital dynamics of a typical object of the Oort Cloud whose self-energy is quite smaller than its putative Rindler energy. By taking a typical comet moving along a highly eccentric and inclined orbit throughout the expected entire extension of the Oort Cloud ( pc), it turns out that the addition of an outward Rindler-like acceleration, that is, for , does not allow bound orbits. Instead, if , the resulting numerically integrated trajectory is limited in space, but it radically differs from the standard Keplerian ellipse. In particular, the heliocentric distance of the comet gets markedly reduced and experiences high-frequency oscillations, its speed is increased, and the overall pattern of the trajectory is quite isotropic. As a consequence, the standard picture of the Oort Cloud is radically altered since its modified orbits are much less sensitive to the disturbing actions of the Galactic tide and nearby passing stars whose effects, in the standard scenario, are responsible for the phenomenology on which our confidence in the existence of the cloud itself is based. The present analysis may be supplemented in future by further statistical Monte Carlo type investigations by randomly varying the initial conditions of the comets.