A generic rich black will usually do for a drop shadow, though you may want to fine tune it based on the color you’re going over (for example, if your shadow falls on a red, then add more magenta and yellow and take away some cyan from your mix)  The baseline that Illustrator offers is 75c 68m 67y 90k – that’s what you’re looking at in the example above.

So, with your object selected, go to Effect -> Stylize -> Drop Shadow… (note as of cs3 or 4 – there are two “stylize” sections.  Look under the top one.)

First, note the Mode setting.  You want Multiply.  Next, click on the square of color to the right of the word Color.  This is where you define the color of the drop shadow. It’ll open up this second dialog:

Note the right column, showing your cmyk breakdown.  All K here.  Adjust it to be a rich black either by manually entering numbers (75c 68m 67y 90k if you want to match my example) or by grabbing that little white circle in the color picker and dragging it all the way into the bottom left corner.  You’ll see the numbers change as you drag it.

Viola.  Rich black.  Hit okay in the color picker and then again in the Drop Shadow dialog and you should be looking at a rich dark shadow instead of the wimpy grey shade you had before.