Why a (banded) black to indigo gradient causes my Dell LCD screen to flicker?

Not my background, but it flickers in Fx5, Chrome 12 and IE8 on my Dell LCD.

If it does, can someone tell me why?

Flickering like this is a commonly known caveat with certain LCD panel designs. You can test which patterns may cause this here. This is a result of the constant voltage switching in the monitor's crystals, which attempt to create a somewhat uniform brightness in the picture you see (this is obviously done at a high frequency so you don't notice).

These voltages are alternated in "interleaving patterns", some of which may cause problems with certain images. Each manufacturer (and even their own sub-products) may use different LCD crystal masks and interleaving patterns, so this flickering effect is caused by different patterns for different monitors.

Since most LCD monitors refresh at around 60 Hz, a full duty cycle for the inversion takes two refresh cycles, causing the flickering at ~30 Hz - a quite noticeable flicker. For a more technical overview of why this occurs, see the Inversion section of this website.

Again, note that this is not a fault of your monitor, this is how all LCD monitors are designed. You can use the test I posted above to determine which pattern type causes this flickering, and avoid their uses if you are a graphics or web designer. Due to image compression when dealing with colour bands (like the question author posted), some colour dithering (which leads to colour banding) can also cause this flickering effect.

This interleaving problem can cause either pixel walk (it looks like the pixels are all moving in a certain direction rapidly) or flickering (flashing at a noticeable frequency) in the image.

I put this exact page on two monitors with different referesh rates - both driven by the same computer. On one you could see the refresh cycle, on the other it did not and was steady, both LCD screens.

I would try to alter the refresh rate (higher?) to see if that diminishes or eliminates the issue. OR Try to match the "optimal" for a particular monitor in color, resolution and refresh.

This seems similar to issues with old monitors and web-safe colors, dithering and the like but not the same thing.