Plantar Fasciitus. Common solution is to rest for a loooong time (as mtulen pointed out). Oddly enough, some runners find that cycling doesn't bother them and turn to it to avoid losing their cardio fitness. Others are warned away from cycling with this problem.
I'm dealing with the same issue right now in my right foot. One question - are you dropping your heels a lot when you push the bigger gears? From what I've read, that and a lack of flexibility in your calves (gastro complex) could be triggering the issue. Stretch those calves and spin more, rather than mashing big gears to avoid making it worse, but if it is plantar fasciitus, you may need to completely rest it.
Arch supports may help - for runners there's a variety of orthotic inserts that support the arch and hold the heel to prevent over-pronation. I'm trying the Specialized BodyGeometry shoes right now, which have a built in varus wedge to keep oddly shaped feet from twisting on the pedal. Normally this (forefoot varus, a twist in the toes) causes knee problems, though, not arch.
Personally, I'm avoiding doctors for now because I don't want to be told to stop cycling