You would normally use the present perfect construction:
I've been assigned another class.
If you want to pin down exactly when in the past you were assigned that class, however, you have to use the past form:
Yesterday I was assigned another class.
If you have not yet been assigned that class but you know that that is going to happen you use the future construction:
Sometime next week I'll be assigned another class.
A note about idiom, in addition to that which jwpat7 provided: assigned to is not incorrect, but at least in US English we usually use just assigned for an additional task or responsibility. Assigned to is more often used when you are moved from one single class or group or unit to a different one:
It's official, I've been assigned to Underwriting.
I'm very disappointed; I've been assigned to Prof. Sartorius' class, and I really wanted to work with Prof. Underhill.
John's no longer in Operations; he's been assigned to Admin.