It's going to be up to the performance of the tuner in the new set, but if your signals are decent enough, it should work fine.
Unfortunately, it's not always the case that a built-in tuner is going to be as good as the converter boxes. The boxes had to meet published specifications for the coupon program but there is no way to know beforehand how good a tuner will be in any given set. If it's a well-known maker, the odds are good that it will have a decent enough tuner. LG, Sony, Samsung, Vizio, and Sharp sets have all done well in my personal experience.