All of the outlets on one of my circuit breakers are no longer working. I changed the breaker out with a new one, but it still doesn't work. What else could the problem be?

Disconnected neutral in the panel, broken conductor somewhere, bad push-wired outlet, loose screw on side-wired outlet, many possibilities. Do you have a non-contact voltage tester? That's the easy way to test for power but no neutral (which I suspect). The other way is to check for voltage between the hot "slot" of the outlet and the GROUND "third prong" socket. Once you establish that there is voltage to the outlet chain you have to go about tracing the open neutral conductor and repairing it. A bit more challenging than an open hot.

EPM

Did you test for 120 coming off the old breaker before you removed it? How about testing the new breaker, one lead on the hot/black and the other lead on the white neutral, what is the reading for both breakers?

Is in the circuit then try finding which outlet the circuit first comes into and if it is an older home there is a good chance its not just the outlets in the circuit on that breaker try to find it and check the connections also look for possible signs of an short or arc that may trip the breaker these may be melting or black spots from electrical burns. Also and this will sound silly but I had an job similar one time and we found a light switch that actually turned on the outlets and they had it fliped off this is common in older homes especially in rooms with out ceiling fixtures, if the room is a living room or a main room odds are it may have something similar. So to really help I would ask the age of home, room, and if you know of switches and all the fixtures in the circuit, is it a 15 amp or 20 amp and how many fixtures are on it?