The setup.exe as part of the download does not recognize 8.1, but you can install by using the .msi in the download folder. The download folder is in your temp folder.. if you downloaded the setup.
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You can also copy the install folder of ArcGIS Explorer desktop and all files from a computer that you have installed on, to an 8.1 machine, set your own shortcut to the e3.exe, and run ArcGIS Explorer desktop on 8.1 as well. This would be the same as the process for running ArcGIS Explorer desktop from a USB drive.
The next version of ArcGIS Explorer desktop setup.exe will install on Windows 8.1
Mark