So, hopefully some of this may help... sorry if I'm blatantly repeating stuff, but maybe it's a good second perspective still.
The flasher relay works mostly like a regular relay, where 12-ish volts across a coil pulls a magnetized contact, which in turn closes a circuit. The difference is that the circuit closure also powers the lamp(s), all while charging/discharging a capacitor. The voltage across that capacitor is what powers the relay coil - a spring holds the contact in the default position until the capacitor reaches a certain voltage, probably around 9V, then the magnet holds it in the other position until it goes back down to a certain lower voltage, probably around 3V.
The resistance in the powered circuit controls the speed at which the capacitor charge changes, but only because it affects the current going through the flasher relay's circuit closure. More current = faster charging/discharging cycle, and less resistance (with approximately constant voltage) = more current.
The real trouble in that last bit is "approximately constant voltage." LED lamps require much more specific (and usually much smaller, though that depends on how many diodes are used in a given lamp) steady currents. More importantly, it's much easier to predict their behavior by applied current than by applied voltage, so LEDs usually need a controller circuit to ensure that the inconsistency of automotive voltage (yes, even with a working regulator) is handled. If they worked for a while and now won't light up at all, it's likely they fried from intermittent overcurrent because the controller wasn't engineered well (e.g., was just some extra resistance stuck in the circuit).
If they are still lighting up, ratmonkey's right on about the digital flasher (I think with '94 you're okay, later years might have trouble with "insufficient current" errors in the BCM). It'd use a discrete timer to control a couple on-board relays or opto-isolators or power transistors, so the timing is independent of the current drawn, and there's no need to put resistors in the circuits (which is a little questionable and maybe dangerous anyway, since they'd have to dissipate a large portion of the same power the original bulbs did, but all as heat instead of mostly as light and some as heat).