North Carolina already has a great rivalry with Duke in the ACC and they meet up twice a year. That's going to stay that way for a long time, but more recently the Tar Heels have been playing some great games outside of the conference like Maryland and others in the ACC/Big Ten Challenge.
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Another team they've played regularly is Kentucky though over the past few years with two meetings in 2011, one in 2013, one in 2014 and they are set to meet in Las Vegas this December. However, with the ACC changing its rules for the amount of conference games it will play, that is likely to put the kibosh on any more series between the two programs.
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UNC coach Roy Williams would love to keep the rivalry going and tried to convince the ACC rule makers to keep the conference schedule down to 18 games, but he had no luck.
"I asked one of those guys [in the NCAA Tournament selection committee] one time, 'Would you rather have North Carolina vs. Kentucky or would you rather have North Carolina and whoever was last in the league?' They said, 'We'd rather have North Carolina/Kentucky.' I said, 'Well, if you have more conference games, you lose some possibilities of having those kind of games.'"
The rules don't go into place until 2019 so the teams will meet again before the series essentially has to get cancelled that year. They could meet in the CBS Sports Classic should each school decide to play in it and it continues getting sponsorship.
You can click here to listen to Roy's explanation, which comes around the 30-minute mark of the podcast he joined with ESPN's Andy Katz.