This sorta thing is under investigation -- and I think will make it into clinical application during our careers. Do a Google scholar search, stuff will pop up. I found this review to get you started.
That said, it's non-trivial, especially in regard to signal and limit of detection -- they picked hydrogen for a reason. MRI's are NOT NMRs -- people are big compared to a 5 mm test-tube, and, even in the most powerful MRIs, actual field strengths experienced by the specimen ***cue creepy music*** are pretty low by comparison. So, getting the LOD down to a range where you can detect clinically relevant concentrations of whatever fluorocarbon you're interested in presents a significant challenge. (Early on, it was even a challenge with hydrogen! The translation of NMR tech to the clinic was not easy, multiply those challenges a million fold with fluorine -- literally, at least looking at the raw signal and concentrations...)
Plus, since it's really damn easy to detect a positron, PET-CT scanners are relatively cheap, and they do the job rather well, from what I understand -- not a huge economic (or even patient-care) incentive to move this research forward.
That said, you're absolutely right -- the hardware IS essentially the same. I'm just a lowly MS1, but I spent 5 years as a chemist in academia (why I know about this crap -- blew way too much of my life designing NMR assays), and my understanding is that the RF coils in MRIs would be readily adaptable to fluorine -- no fundamental hardware or software limitations. That's not the challenge here. It's the low field strength and low F concentration that's the issue -- gotta get that signal up, and the LOD down. Which is not easy.
That said, I imagine this will happen someday (soonish), if the economics allow...