Previously Proposed Explanations for the Anomaly
Same Effect, Different Spacecraft
Many of the explanations for the slowdown depended on the specific spacecraft, for instance, on a gas leak or a malfunctioning thruster from the particular design of the particular spacecraft. However, the Pioneer anomaly has been observed by all such spacecraft, so having different explanations for the individual spacecraft for a single effect doesn't make a whole lot of sense.
Some people had put forth that there may have been some slight, constant errors made in measuring and computing the Pioneer anomaly. Computing errors don't seem too likely considering that people have checked and rechecked these calculations over the years with this already in mind. However, the possibility existed that some of the Pioneer effect was due to measurement error. The various spacecraft are all difficult to track in their different ways, on top of their great distance and small size. Acquiring position and velocity data to the degree of accuracy that the Pioneer effect requires isn't possible all the time for all the spacecraft because of noise in the transmission and other little things.
There were a number of explanations that insisted the Pioneer effect is really just real deceleration that scientists didn't account for in their predictions. Some of these had more merit than others.
A common explanation focused on the belief that other gravitational influences on the spacecraft that we did not consider or know about, perhaps an undiscovered pocket of dark matter or a hefty new planet, caused the effect. However, seeing that other far-off objects, such as the outer planets, do not seem to be under the influence of any mysterious, unknown gravitational forces, this theory seemed unlikely.
Deceleration from drag by the interplanetary medium was another proposal. However, the measured density of this dust, solar wind, cosmic rays and the like was simply too small to make as big of an effect as the discrepancies required. Drag from radiation pressure was a similar explanation — and one that was also too small to account for much of the Pioneer anomaly. The pressure of sunlight is simply too small at so great a distance. However, some estimates suggested that as much as a third of the Pioneer anomaly might be due to radiation pressure and heating.
Hubble Constant & the Expansion of the Universe
The precise magnitude of the Pioneer effect happens to be rather close to the product of the speed of light and the Hubble Constant, by which the expansion of the universe is measured. This led to some speculation that the Pioneer effect might have something to do with the expansion of the universe. However, this idea could be dismissed when one observed that our local gravitationally bound objects—such as all objects in the solar system—are not included in this effect, as proved by theory and by direct measurement.
MOND: Modified Newtonian Physics
Some people had proposed, using everything from the Pioneer anomaly to reinterpretations of dark matter as supporting evidence, a few tiny alterations to the theory of gravity. MOND, or Modified Newtonian Dynamics, proposed that gravity changed on very low accelerations. This may also be interpreted as a modification to theories of inertia, applied to the Pioneer effect.
This final theory, the tired light explanation, was a little more "out there" than the others, but impossible to either accept or refute. This theory suggested that light will simply lose energy over time. For the Pioneer anomaly, this just results in a case of miscommunication between Earth and our spacecraft. For the rest of the universe... well, let's just say that Big Bang didn't happen quite according to mainstream theory. Tired light was not a commonly accepted theory, however, and had many errors, as pointed out here.