Not as good as Oly for sure. I'm using em5 n XE2 with latest firmware, the xe2 often misfocus, even with 18-55 which is supposed to be very good in AF in fuji lineup. XT1 should share a similar AF system with Xe2
Hi , I've the EM1 and XT1 and i have used the EM5 and XE2 before this.
XT1 vs XE2 first: XT1 is faster, especially on the primes. It's noticeably. Using the 18-55 and 27/2.8 seems to be the fastest on both bodies in AF.
Now XT1 vs EM1:
AF speed: EM1 has the edge but this is dependent on lens too.
AF accuracy : EM1 has the edge.
But remember EM1 is the top tier model from a very mature MFT , compared to the X-Trans series I still see as relatively young.
That is not to say XT1 is slow or mis-focus often. Even when doing street with the XE2 and 35/1.4 I don't get much missed shots but if u compare the EM1 + 25/1.8 vs the XE2 + 35/1.4 of course EM1 will be significantly faster for the same FL.
Using the EM1 , 12-40/2.8 Is a wonderful combination but I can't compare this to Fuji as Fuji hasn't release their 16-55/2.8 yet.
I'll instead say it as these 2 are different beasts. Most users probably feel this way as the XT1 or XE series needs more training in using them to get used to the Fuji system quirks.
Both systems will be able to give anyone what they want , Olympus for the Oly colors and the high end performance / stability everyone now looks up to it for; Fuji for the Fuji film-like colors and the good glass everyone loves it for.
Other brands better start catching up in the Mirrorless arena.

for example I'm not even interested in the RX Mark III when looking at the specs it's supposed to be a killer.