It has certainly been possible to get a job flying B737s straight from training. The only operator of B767/777s that did take low-hour pilots was BA, but as far as I know the largest type they'd put newly-graduated cadets on is short-haul B757s. A couple of IT operators have also put low-hour pilots on B757s in the recent past.
However, that was all before Sep 11th! Right now, there are very, very few jobs out there for fATPLs. I'm sure the situation will be much better in four years' time, but I wouldn't be in such a rush to run before you can walk.