This thread has been latent for a while already, but I´m going to stir it a little, since I think there is more to be said on the subject than is stated here.
I´m using the same version ArcGIS and I could´nt get the Clementini-trick posted by mapBaker to work on my system. I kept getting the same results as "intesects" (don´know why yet, mystery still to be resolved), but thats not what is asked for in the question. ArcGIS does however have a method for selecting called "have their centroid in the source layer", which should work in most cases of this particular task.
However, you could have a situation where the centroid of a block group falls outside the border of a county. In this case that particular block group won´t be selected and your selection will not be complete. When this is the case you´ll have to switch tools and hop on to QGIS, which has a solution for this problem. In this article by Heikki Vesanto you´ll find the workflow laid out. This functionality is nowadays available as a QGIS plugin, which makes the task even much more simple.