Is it possible that the polarity is reversed at your outlet? Are the hot and neutral switched in the service panel (neutral connected to the breaker, hot connected to the neutral bus). If so, that could potentially cook a device that is sensitive to polarity, and maybe their reference to "grounded neutral" was an unclear reference to the grounded conductor being connected to the wrong end of the circuit inside your device?
If the device had a two-prong outlet, as @maple_shart said, then grounding is not your issue, presuming that's the correct plug for the circuitry in the device (is it UL listed?).
If the plug is polarized (one prong is wider than the other), then the device is sensitive to the polarity of the electrical supply in some way, possibly because it only has a single-pole switch making it unsafe for humans if the neutral is switched instead of the "hot" conductor.
If the plug is not polarized (both prongs are the same width), then the device should not be sensitive to polarity.
Every correctly-wired neutral is a grounded neutral. The neutral is the grounded conductor. Even if you're talking about a subpanel with the grounded (neutral) and grounding buses unbonded, the neutral and grounding buses are (supposed to be) bonded in the main panel, and the grounding bus in that panel is supposed to be bonded to grounding rods and/or water pipes (as appropriate, for anything new it should be a couple of grounding rods, and also your water pipes if they are metal, so that they don't end up accidentally energized with no safe current path). But even if the neutral isn't grounded at your main panel, it's grounded at the power company's transformer (that bare copper wire running down the pole from the transformer to a ground rod at the base of the pole).