We never had an Ohio State / Michigan rivalry. I think Syracuse was the closest, but I may be biased because I knew a lot of people that went there. I think our rivals were cyclical, depending on how good we were and how good another team was over a period of a few years.
A true rivalry would survive a conference change. I think there are geographical rivalries that survived the Big East break up. BC and Providence come to mind. PC was never a rival to me because I'm a NJ public school guy. I never knew anyone that went to Providence ( had friends at BC). But clearly, Providence always hated UConn and I always loved that many at UConn didn't care about Providence.
Pitt was a rival for a while, and there were certainly some St. John's angst when we would take over their building with Jarvis there, but that was circumstantial since we all stopped caring the moment they both left the Big East (yes, you too St. John's). Syracuse is the only one that I think we can call a real rivalry since the loathing still seems in place.
Who will be our rival? I don't think we have enough in common with any Big East institution to have a permanent rivalry. WHEN we become a perennial top 3 finisher, they'll hate us, and we'll hate the others in our stratosphere, but I think we're just too different to care about another Big East school every year. Providence will always hate us though and I still don't care about them (other than now having an obnoxious neighbor that went there).