Dunno about that. The quad-core AMD Opteron-powered Linux server behind my desk is very fast (about 22,000 BogoMips), and it uses interrupts...lots of them.
It depends on what the system is expected to do. The problem with not having an interrupt system is that asynchronous I/O can become quite inefficient, as the MPU has to constantly check all possible I/O sources for activity, which reduces the time available to execute other tasks, such as run a user application. Also, you'd have to be constantly saving the state of the MPU as it went from device to device looking for activity to service. I could see where it could become a programming nightmare.
If interrupts are involved, polling doesn't have to occur at all until something demands attention and the ISR can be designed to preserve the MPU state so the main task continues without a hitch after the interrupt has been processed. I think you'd find it cumbersome to work with a general purpose computer that didn't have a means of reacting asynchronously to external events (e.g., typing on a keyboard). That is where interrupts can be your friend.