I had that same problem also. I also changed Transmission to run as root.
I did try to change the permissions and owner of the folder i was writing to, and it didn't work. I'm not quite sure why.
It runs fine now as root, so I'll just keep it that way.
Funny enough i had the similar problem with my external ntfs drive.
External file systems when mounted can only be written by too by their owner (root usually)
Any way i was able to mount as root and write to the file system using mount device mount-point but i was unable to write as the user transmission
It turned out i needed to mount my external hd with fstab i needed to specify the mount type as ntfs-3g
mount -t ntfs-3g /dev/sdb1 /media/harddrive
I recommend that you use fstab to mount external hard drives that you connect to the plug frequently