What is a 'grounded neutral' and how bad is it?
That explanation doesn't really make any sense to me personally.  The neutral bus bar in your main panel should be bonded to the ground bus bar so in effect, all neutral wires in your house are grounded.

The third prong in a typical three prong outlet is known as the equipment ground.  This is a safety feature that should cause your breaker to trip if an electrical fault inside of an appliance causes the metal body of the appliance to become energized.  The equipment ground protection however is a safety feature for YOU so that you won't be electrocuted in the case of an appliance fault.  It doesn't and shouldn't make any difference at all do the proper functioning of an appliance.

If an appliance you (recently purchased, older appliances may not have the same quality standards!) purchased has only a two prong plug, then it is essentially assured to not be an electrocution hazard in the case of an electrical fault, or it is not capable of having such a fault.

So basically what I am saying, the appliance you are talking about would never have been grounded anyway if it has a two prong plug.  Grounding has nothing to do with the failure of your appliance.

You must have misheard because perhaps the tech support person meant to say "Ungrounded Neutral" which potentially could be a problem for sensitive electronics.  The neutral (connected to the center tap in the main panel) can and does carry an electrical charge, so the neutral bus bar should be grounded to the outside through the use of a grounding rod to bring Neutral to Earth Ground.

If your home was built in 1959 then it is possible that the grounding rod was never placed deep enough into the earth to provide proper grounding, or perhaps it rusted away over time.  How far the grounding rod needs to go depends highly on the region you live in too. I used to live in a home built in 1958 and had a similar problem. A qualified electrician would be able to fix this for you, or an adventurous DIY'er.