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The vibration goes away once you get out of that range, I'm inclined to think it's either the engine or transmission because this happens no matter which gear I'm in.

Is the vibration felt through the steering wheel or through your seat?

Its probably a good time to change your engine mounts, trans mounts, guibo and CSB.

Guibo? Center bearing?

What's your exhaust system like? Could you be feeling a resonance in the exhaust system? Engine and exhaust are the only things that are RPM sensitive, tranny thru diff are speed sensitive.

I was mistaken, it's only noticeable in 3rd gear. The vibration is all around, I can feel it through the steering wheel too, but really the whole chassis has a slight shake to it.

The previous owner took the exhaust system from a 6 series.

The vibration started after I did a donut in a parking lot >_<

Like Ken & Jody suggested, check your guibo with a flashlight.
You're looking for signs of tears in the rubber.

I took a look at the guibo and it looks new. Is there a way to check the center bearings without having to remove the driveshaft?

My experience with center bearings is when they wear there is a vibration (or a knocking against the driveshaft tunnel with severe wear) upon takeoff from a stop. That is when the most torque is applied on the shaft, causing it to bend slightly. The vibration goes away after a second or two when the driveshaft straightens out.

Actually, the vibration happens at 25-30 mph no matter which gear I'm in.

Depending on the frequency of the vibration, it might even be a tire / wheel problem.

Maybe it's worth to check your engine and gearbox mounts - on my E32 the gearbox moved out of line once after making an enthusiast left turn. Besides a short moment of scary noises I also got a vibration.