Many inkjet printers actually even use (and thus: require) colour when printing in greyscale. If the colour cartridge is empty, such printers might refuse to print in greyscale, and thus will not print images. (And those might even not print black text when selecting greyscale.)
When you don't need colour, get a black and white printer instead.
Odd things have been reported. Like for some brands printing in black and white using best quality avoids using the colour cartridge, while selecting some economy mode would actually use colour:
Is there a driver setting [..] to print photos using the black cartridges for the printer?
Until the iP4500, it was enough to check "Grayscale" and select "plain paper" and "high quality".
And, things might be much more complicated, like for the Canon MP760 and the iP4000, which use a large BCI-3e pigment black cartridge, and smaller BCI-6 dye inks including yet another black (and yellow, cyan and magenta):
The truth is that the black pigment ink is [almost] always used on plain paper--never on photo paper and the dye black ink is always used on photo paper--[almost] never on plain paper.
The exception is when doing borderless printing which is not recommended on plain paper, but if you do it, the printer will use the dye black ink instead of the pigment black ink.
Pretty simple, but there is a complication involving duplex printing. When doing duplex printing, the black pigment ink is cut to about one half of the intensity that it would normally print. Then, to make up for this decrease, the magenta and cyan inks are printed in the same area as the pigment black. The yellow and dye black are never added to the pigment black to make up the difference. NOTE: Recent tests have shown that this last statement is only true for 100% black. In fact, yellow, along with magenta and cyan, is blended for some shades of grey. Also, it doesn't matter whether the duplex printing is done automatically or manually, or whether you are doing booklet printing. And, in all cases, we are talking about duplex printing on plain paper.
(As a side note: some laser printers are known to count colour usage even when printing in black and white.)