I've been experiencing foot cramps during and after some of my longer rides (which for me means approximately 20+ miles). Any idea what's causing that and how I can remedy it? If the pedals and/or shoes matter, I'm using plain old pedals with no toe clips, and I just wear running shoes.

Based on my experience, 2 reasons that caused me to have foot cramps.
The first was when I was just starting to get into bicycling and my foot and leg muscles weren't in shape yet. I'd get awful cramps until about 7 months of bicycling. If you're doing 20 miles regularly this break in period may be shorter. Try squatting down afew times and stretch the foot muscles before a long ride.

The second reason happened last Sunday when I set the saddle height to my hybrid too high and had to stretch on the downstroke of the pedal. I haden't ridden my hybrid for a long while and was messing with the saddle height in the middle of the ride and had bad foot cramps on the way home. Adjusting the saddle height cured that. Hope this helps.
Ernest

sore feet generally means you are in too high of a gear (pushing too hard). try spinning faster and see if it is easier on your feet.

Are the cramps in the sole of your foot?

If so, it might be "Hot Foot" which is related to the metatarsals not having enough room, fixes include wider shoes, looser straps over the ball of the foot, moving the cleat rearward, different insoles, orthotics, etc.

For me, most foot and calf cramping has to do with improper hydration. Make sure you drink enough several hours pre ride.

There is a reason cycling shoes have a very stiff sole. This keeps the foot from flexing, which over time can lead to fatigue and cramps. If this is the cause riding with your toes on the pedals as opposed to the balls of your feet tends to make the problem worse. The hydration and fit issues raised earlier are also valid and could be a compounding factor or the main cause.

I'll throw this out there..

You're not diabetic are you? Problems will usually manifest themselves in the extremeties, particularly the feet first.

Is your family Italian somewhere? This could be caused by the onset of Gout. Otherwise known as the Mediterranean Familial Fever, gout is passed up through the genetic code from people who were from - sometimes way back - the Italian coastal regions.

Anyone up for a friendly game of Trivial Pursuit? For money?