The Doppler shift is a phenomenon that occurs when an object is moving towars or away from an observer. You can observe this effect by watching a police car drive by on the road with its siren turned on; as the car approaches you, the pitch of the siren grows steadily higher, peaking when the car reaches you, then the pitch grows steadily lower as the car moves away from you. This is a consequence of the fact that sound is carried by waves; as the car approaches, the sound waves arw "squished" (a simplified explanation); as the car recedes, the waves are "stretched" (again, simplified). The same effect occurs with electromagnetic waves, such as light.
To measure the Doppler Shift of stars, we look at how the stars' color varies. Remember that with sound waves, the pitch grows higher as an object approaches and lower as it recedes? With light waves, the color of an object moves towards the blue end of the visible light spectrum as the object approaches an observer (a "blue shift") and towards the red end as the object recedes (a "red shift"). When we measure the Doppler effect on nearby stars, we see that the amount and direction of the shift changes with a regular period. This is easily explained by a heliocentric model of the solar system; as the Earth approaches the part of its orbit that is nearest a particular star, the star has a blue shift, and as the Earth moves away from the star, the star has a red shift. However, if we are being accelerated at 9.8m/s/s, we should be approaching lightspeed on an asymptotic curve, and be moving very, very nearly at lightspeed relative to external stars. Therefore, we should not see a periodic change in shifts, but rather a constant shift on all stars, at least until we pass by them- stars in front of our acceleration should be exceedingly heavily blue shifted, and stars behind us should be exceedingly heavily red shifted.
Another problem also arises; we should be racing past the nearby stars, unless of course FE posits that the entire galaxy is being accelerated. However, if the galaxy is indeed being accelerated we should measure massive blueshifts on galaxies in half the sky (the part that is in front of us) and massive redshifts on the other half. Instead, we measure large (but much smaller than would be expected if we were moving near lightspeed) redshifts on the part of the great majority of galaxies in the universe. Unless, of course, the entire universe is being accelerated, but then the question arises, accelerated relative to what?