White balance tools (like those in Lightroom or Photoshop) will tell you the color temperature of the neutral card essentially directly, based on the color temperature data in the RAW file.

More theoretically, if you're shooting with a known color temperature (even approximately known), then the color of the neutral target is directly related to the color temperature of the lighting. Doing this precisely has several big catches (e.g., it doesn't really apply to fluorescent or mixed lighting, and understanding color theory is a prerequisite), but doing it relatively is eminently practical:

• if the card is bluish, it's lit with a higher color temperature than your setting.
• if the card is reddish, it's lit with a lower color temperature than your setting.

With practice, the card isn't needed; you can see the relevant color casts in any neutral object, and particularly in shadows.

Personally, I think the best advice is to try to memorize the approximate color temperatures of typical situations, like this chart from Cambridge in Colour (the full article is worth a read):

• 1000-2000K – Candlelight
• 2500-3500K – Tungsten Bulb (household variety)
• 3000-4000K – Sunrise/Sunset (clear sky)
• 4000-5000K – Fluorescent Lamps1
• 5000-5500K – Electronic Flash
• 5000-6500K – Daylight with Clear Sky (sun overhead)2
• 6500-8000K – Moderately Overcast Sky
• 9000-10000K – Shade or Heavily Overcast Sky

1 Fluorescent lamps aren't standard light sources, and many variations exist, so this should be considered a very approximate figure.
2 There's an old joke about the noon sky outside of Kodak's offices being exactly 5500K.