Um, an automatic is a mechanical watch. I'm going to assume that you mean hand-wound watches as opposed to self-winding (automatic).
Anyway, what you heard is B.S. That oil's not going anywhere, even if the watch isn't running. Why? In a film of oil that thin, the force of gravity is negligible compared to the cohesive force between the metal and the oil. The amount of oil used in a watch movement is tiny, and it's not like a car engine which circulates its oil.
Having said that, it doesn't hurt to run a watch once in a while, but it doesn't help much either. I've never heard of modern oils used in movements "gumming up" from lack of motion; they are extremely good for their application.
Also, a handwound watch doesn't necessarily have fewer moving parts. Depending on which complications it has, it can have way more parts than an auto. Check out Patek Philippe's Caliber 89 (ok, that's a pocket watch) or anything with a minute repeater, tourbillon, rattrappante, or in some cases all 3 in the same watch.
The rotor and winding mechanism of an auto aren't very sensitive to lubrication. The escapement is, and this is found in all mechanical watches. So anything that applies to autos as far as lube is concerned also goes for handwounds.
Anyway, don't sweat it if you leave a watch sitting around for a long time. Even if the oil was unevenly distributed on the bearing surface due to creep from sitting around for a year, it's still confined within the proper place. It's not like it's going to end up on the bottom of the case or something. So as soon as it starts moving again, it will be evenly distributed.
Of course, nobody who owns a mechanical watch is going to let it sit around that long anyway, they're just way too cool to play with...