I plan to post a summary of responses to this as soon as I have working
code, which I will also include.  The intersection of 3 planes method
looks best, but my implementation based on a short article in 
Graphics Gems I doesn't work.  I may be misinterpreting, of course.

I had avoided the simultaneous solution of the plane equations in favor
of dot and cross products, but the former may actually be better.  In either
case a matrix determinant needs to be computed (implicitly in the solution
of linear equations).

To get the planes, I was taking the midpoint of the line from, say,
P1 to P2, and setting the normal as the "normalized" vector from P1 to P2.
These just plugged into the formula in Graphics Gems.  HOwever, the resulting
center point is only occasionally equidistant from all 4 of my test points
(for different tests).  My matrix/vector math is very rusty, but it looks like
I need to verify the formula, or use the simultaneous equation solution, which
will require bringing in another routine I don't have (but should be easy to
find).

Another method is to first find the center of the circle defined by 2 sets
of 3 points, and intersecting the normals from there.  This would also define
the circle center.  However, small numerical imprecisions would make the
lines not intersect.  Supposedly 3 planes HAVE to intersect in a unique
point if they are not parallel.

Ed

Thanks to all who answered so far.