Wow, that worked a lot easier that I thought it would. You can now pass a querystring of ?return.ObjectTpe=Property1,Property2,...
Here is a sample for getting the diameter from two circles
curl -H "api_token: steve@mcneel.com" -H "Content-Type: application/json"
-d '[[{"X":1.0,"Y":2.0,"Z":3.0},12],[{"X":1.0,"Y":2.0,"Z":4.0},30]]'
"https://compute.rhino3d.com/Rhino/Geometry/Circle/New?multiple=true&return.Circle=Diameter"
Here is a sample for getting Vertices and Faces from a computed mesh (note, I need to clean up JSON serialization for bounding boxes, planes, circles, arcs, and spheres still). This one is a sphere so it is still a little ugly.
curl -H "api_token: steve@mcneel.com" -H "Content-Type: application/json"
-d '[{"IsValid":true,"BoundingBox":{"IsValid":true,"Min":{"X":-11.0,"Y":-10.0,"Z":-9.0},"Max":{"X":13.0,"Y":14.0,"Z":15.0},"Center":
{"X":1.0,"Y":2.0,"Z":3.0},"Area":3456.0,"Volume":13824.0,"Diagonal":
{"X":24.0,"Y":24.0,"Z":24.0}},"Diameter":24.0,"Radius":12.0,"EquitorialPlane":{"Origin":
{"X":1.0,"Y":2.0,"Z":3.0},"XAxis":{"X":1.0,"Y":0.0,"Z":0.0},"YAxis":{"X":0.0,"Y":1.0,"Z":0.0},"ZAxis":
{"X":0.0,"Y":0.0,"Z":1.0},"Normal":{"X":0.0,"Y":0.0,"Z":1.0}},"EquatorialPlane":{"Origin":
{"X":1.0,"Y":2.0,"Z":3.0},"XAxis":{"X":1.0,"Y":0.0,"Z":0.0},"YAxis":{"X":0.0,"Y":1.0,"Z":0.0},"ZAxis":
{"X":0.0,"Y":0.0,"Z":1.0},"Normal":{"X":0.0,"Y":0.0,"Z":1.0}},"Center":
{"X":1.0,"Y":2.0,"Z":3.0},"NorthPole":{"X":1.0000000000000007,"Y":2.0,"Z":15.0},"SouthPole":
{"X":1.0000000000000007,"Y":2.0,"Z":-9.0}},10,10]'
"https://compute.rhino3d.com/Rhino/Geometry/Mesh/CreateFromSphere?return.Mesh=Vertices,Faces"