I guess my biggest fear about Cassandra's Tears as a GM hook is that it could be abused way too easily. I once played in a D&D campaign that became known as "The Green Arrow Inn Detective Agency Campaign", because every time we finished an adventure or story arc the GM used his favorite bartender to introduce the new one. I wouldn't want - as a player or as a GM - the campaign to end up being "Mrs. Cassandra and her Detective Agency", y'know?
It's a wonderful tool, and something with a lot of potential. I guess I just worry about the ease of mis-using it.
I'd tend more towards using it for less... Earth-shattering things. That, or just more vague things.
Or, you could leave them with some nasty, longer-term ones. A prediction is an aspect on the world - it doesn't have to be an aspect resolved immediately. How long did Harry have "Die Alone" (a death curse, but still) hanging over his head before it was resolved? If it even has now?
I would try to get into the habit of having "tiers" of predictions. One or two big, scary, multi-plot-arc ominous predictions that hang over their heads like the sword of Damocles ("New York will burn because of you."); these also have the advantage of being potential major RP points for the character... Since, by definition, people have trouble believing those with the gift. Maybe 3-4 mid-tier predictions, relative to the plot arc; these may be minor, or they may give a key clue or motivation. And lastly, depending on how long you guys play and as long as you're careful not to make the entire campaign about this, however many minor predictions you're comfortable with; I may even say one per scene, if it works for you guys.
I mean, you can't be giving out fate points like candy, but for a relatively minor thing your players may agree that it's not relevant - after all, the system is pretty specific that things which aren't significant should be assumed and moved past. And the fate points should be going both ways, anyway - the player should be compelling it (or attempting to) as well as you.
And on top of it, never forget the confusing nature of prophecy. The prophecy "New York will burn because of your actions" sounds incredibly ominous, but has miles and miles of leeway. I'd almost dance with glee to dump this aspect onto a story I was running. Does it refer to the state or the city? Is it literal or figurative? Could it refer to a model of the city, in the way Harry had a model of Chicago? Could it refer to the government - the political power structure? Could it refer to the players igniting a feud between supernatural factions, starting what amounts to a turf war? How about the players getting into a fight in a warehouse and starting a major fire on the docks? Could the city itself suffer a major catastrophe, enough to change the world, that the players are directly involved in? It sounds bad, but what if fulfilling this prophecy is the result of avoiding something worse? Don't belittle the prediction, but it never has to be obvious.