Change your motor mounts first. Then fgure out the rest later.

I had the same symptoms with both my '85 300D and my '82 300CD.
The cause turned out to be worn suspension ball joints.  Is there
uneven tread wear on either of your front tires?  If so, the ball
joints are the likely culprit.

Geoff

i had a similar prob, the flex ehaust mount to the transmission had
came off.

It could be any number of things but I'd be very surprised if changing the
motor mounts fixed it. At the age the car is now it probably needs new:
tie rods, idler arm and all soft rubber parts in the front suspension
renewed. If you plan on keeping the car long to medium term bite the
bullet anbd repalce all the wear items. If you just want tofix this
noice which frankly is penny wise and pound foolish then you should
have a competane mechanic inspect it. It's toug hto diagnose this
over the wire modulo a lucky guess.

I replaced the left and right motor mounts, and the grinding type noise
as well as the vibration when the car is in reverse are still present.  
It also seems as if the grinding noise may be easier to reproduce now, as
I don't think I was turning right when I heard it.  The vehicle also
seems to be just generally rougher running than it used to be.

The work that has been done on the car fairly recently includes new motor
mounts, replacing tie rod ends on passenger side, new lower ball joints
on both sides, new wheel bearings on both sides, new strut rod bushings
on both sides, and a new center carrier and carrier bearing.

I had the front end inspected by a respected Mercedes mechanic perhaps
7,000-10,000 miles ago, and he instructed me to do all the mentioned
repairs (with the exception of the tie rod ends and motor mounts).

I'll look into the suggestions already provided, but with this new
information, does anyone have any further suggestions?

Thanks

I drove the car some more and noticed that the noise seems to have gotten
worse after changing the motor mounts.  The grinding noise seems to now be
present when the wheel is centered or turned to the right (but goes away or
at least gets quieter when turning to the left), but only if the car is
going above about 45 or 50 mph.  This sounds like tires to me, except that
the noise shows up when I let my foot off of the pedal, too.

Additionally, the vehicle is loud and vibrates a lot when it is idling.  
Normally I would think this is motor mounts, but, as I said, they've just
been replaced (2 days ago).  The vibration gets worse when the vehicle is
shifted into reverse and the brake is held in.

The underside of the car was inspected.  The exhaust flex mount on the
transmission seemed fine, and best I can tell the tie rod ends were fine on
the side that hasn't already been replaced.  The only thing that was
obviously broken or worn was a rubber hanger on the muffler.

I'll look again to see if I see grinding anywhere...

I took it to a Mercedes mechanic.  He immediately thought that there was a
motor mount problem (although they had just been replaced a few days
earlier).  It turns out that there was something (I think it was part of
one of the A/C hoses) pressed against the motor and the chassis, and this
seemed to transfer the vibration from the motor to the chassis.  This was
bent back out of the way and the problem seems to have disappeared for the
time being.