You can have two issues that contribute to that. The most significant is external vibration getting into the table. You can try isolating the turntable by using several different methods. One would be to place the table on a very heavy, stable stand. The other would be (perhaps in addition to that) to add a form of isolation such as a sand base or an air base underneath. A wall mount will help in case of a springy floor.
A more troubling issue can be resonance that is created due to the improper matching of cartridge to arm. Certain cartridges must be used with certain arms and you need to know a few specs and apply a formula to choose the best mates. The arms have an "effective mass" figure and the cartridges have both "net weight or mass" and "compliance" which must be used in the formula. Getting an arm/cartridge system that does not have a problematic resonant frequency is important and can prevent such behavior.
The last thing that can happen is acoustic feedback, but it sounds like your problem is related to one or both of the above issues as it occurs without any signal. Accoustic feedback occurs when a system is played loudly and the speakers cause vibrations that effect the table and get reproduced through the speakers again causing an escallating loop.
-Bill

Uptown seems to have nailed it.

I don't know dick about turntables, but I was browsing through an old issue of Stereophile and ran across an interesting review of a Grado ZTE+1 cartridge. Quoting a portion of the review..."Will hum if used with older AR decks; lack of suspension damping can lead to woofer pumping, even flutter, with high or even medium mass arms."

You can check for external vibration by placing your tonearm on a record with the turntable still. You may have to unplug the power to do this. If no vibration, then you have a arm problem and isolation of the table won't help. The other factor is the freq response of your system. You may need to use a subsonic filter because the motion is probably due to record warp which you cannot adequately eliminate. This is a fact of life in LP. Adjustment of the effective mass of the tonearm can mitigate the situation.
A solution which is surprisingly effective is the damping brush on the cartridge like the shure M97 and the V15 (discontinued).

I may be totally dating myself here, but this issue used to bring up the use of subsonic filters in phono preamps.

Are these no longer part of hi fi?

Quote:
I may be totally dating myself here, but this issue used to bring up the use of subsonic filters in phono preamps. Are these no longer part of hi fi?

It's difficult to implement a high-pass rumble filter that doesn't also have audible effects at higher frequencies. It's best to deal with subsonic rubbish by using a turntable with low rumble and carefully matching the tonearm effective mass+cartridge mass to the cartridge-suspension compliance to avoid amplifying warp-induced spuriae.

John Atkinson
Editor, Stereophile

I measured the rumble on my MMF5 with a sound card, and on a spectrogram it is readily visible as high as 35Hz and goes all the way down to DC. Furthermore, some LPs appear to have a legitimate signal down to 8hz, which is below the resonant frequency of most tables. So it's not like you can draw a line and say "everything under this frequency is noise/rumble". The only decent way to take care of it is to use a table which rumbles less.

That said there is very little in the way of objective information out there on the rumble characteristics of different turntables, beyond the obvious requirements for cart matching. For now, use your ears, I guess. (And a good sub.)