This is what we did last year - we booked the restaurant for the first night before we left, although unless you really want Cafe Mickey you probably won't need to do this. Then after a wee look at all the restaurants, we booked restaurants for the rest of the stay at the concierge in HNY on our second day. This was really easy, the man at concierge was very helpful and wrote down all our booking times so we wouldn't forget.
This year I've been a bit more obsessive with my planning and have booked everything in advance - not so much out of worry that we wouldn't get in as from a love of organising