I don't have any experience with a problem like this, but I have an idea which may help.
Regular bulbs don't care which way positive is coming from. You can't get it wrong. LED's DO care though. The long wire is the positive and the short wire is the negative on a single LED which would go into a bread board.
My point is that if it is as one pointed out that the off bulbs actually ground the other bulbs, but those LED's or the previous LED's were backwards, then they could possibly all turn on. Maybe on one side of the bike switch the wires and see if that works as the polarity of the LED, which is also a diode, that's what the D in LED means.
Sounds good, but no. The LED bulbs are connected only one way, and that is with the positive on the center conductor (or conductors, in the case of the front bulbs), and negative on the outer shell, assuming 1156 or 1157-style bulbs. It's just that very little current is required to make one glow, less than what the flasher indicator will conduct. The 'off' side will be dimmer than the 'on' side, but not a lot, even if the indicator has been replaced with a LED too. The solution is the pair of diodes as illustrated above, or putting two of the filament bulbs back in. A typical turn signal filament lamp is ~21W when on, which is about 1.6A at ~7.5 Ohms, but that's hot. When cold, the filament is about 1/10 the resistance, or 0.75 Ohms, close enough to look like ground to the little 5W bulb used for most indicators. Now, replace that filament with an LED, which doesn't conduct until the Voltage rises over about 2V, at which point it starts to glow; it doesn't take much power through the indicator, even a LED one, to make the off side light up, because full brightness is reached at about 0.25A, probably less, and the indicator probably conducts about 0.1A.