I remember seeing somewhere that Eyefinity doesn't work with crossfire yet...OK - mine arrived today as well.
Have had a little while to play with it - on my Win 7 x64, 5970 BE, i920, 3x 22' Monitor system ....
Have issues getting all three monitors to run together - any two of three is fine (2x DVi or 1xDVI and 1xDP via adaptor) but not all three.
Windows doesn't seem to like enabling the DP monitor either - get an 'unable to save settings' message from Windows when I try.
Out of time for the moment - but it seems it is a maybe rather than a cert.
Have you tryed disabling crossfire on your 5970 ...