It sounds like something is grinding underneath my car when I slowly accelerate, could it by the heat shield bumping against something?

It's not constant so something must have come loose and brushes up against the driveshaft??

 

Possible the driveshaft supporter. Mine came on at 25 mph. Replaced the part and the noise was gone ($300+)

 

I had a heat shield making a similar noise, jack her up and hit all of the heat sheilds with the handle of a screw driver until you find the problem.

 

Mine was an exhaust pipe bracket. Shake the pipes back and forth and if they're shakeable, tighten the mounts. Problem solved.

 

I also had a loose heatshield clamp that made a rattle sound. Grinding noise usually caused by moving parts.

 

I noticed the sound today at idle...

 

As the cold weather is moving in, you'll hear more of the unusual noise especially when driving on a rough road. I am planning to bring my car to Jiffy Lube where they claim they don't just change oil, but also lube (hopefully the suspension components). For older cars, you'll hear strange noises inside the cabin too. That's the disavantage of Lexus cars, it's too quiet and you can detect everything little unusual noise

When I drive my weekend car, I would care less about any noise in the cabin, unless it's coming from the engine.

 

There was a crack in the heatshield, a spot weld and now I am as good as new!!