Why won't my dwarves pick up a pick and go mine?

You need two things in addition to the picks.

• Dwarves with the mining labor enabled, as explained here.
  • The dwarves might be busy with other things, try turning off other labors.
• An area designated for digging.
  • You can designate an area for digging by pressing d, d and selecting an area to be mined. This is easiest if you have an exposed hillside on the map. If not you may need to go down a z level and designate ramps up (d, r) next to your mining area.
  • The dwarves must be able to path to this digging area. If it is separated by z-levels, you must add stairs or ramps.

Are you using Dwarf Therapist by any chance? That had a bug where enabling Mining via DT didn't get the dwarves to actually do anything, and you had to disable and re-enable Mining via the game's interface (as in C. Ross's answer).

Another thing to check is whether the picks are not, for some strange reason, forbidden.