No truth to it at all.
As others have mentioned, there were a number of uses for wood-bulleted cartridges, but not for primary combat use, and not by just the Germans, either.
This practice also continued AFTER WW II for a number of nations, as well.
Here's what Jean Huon has to say about different wood-bulleted cartridges in 7.92x57...
"Blank Model 1888 for rifle, "Platzpatrone 88 Gewehr" -- hyollow wooden bullet painted red, propellant covered with felt wad."
"Bland for machine guns, "Platzpatrone MG" -- hollow wooden bullet painted blue and heavier loads than for the rifle blanks."
"Blank Model 1933 "Platzpatrone 33" -- hollow wooden bullet painted red, pink, blie, or violet, 1 gr. (gram, not grain) load of special blank powder, with a felt wad" (foot note says that One or two knurled rings around the case body indicates it has been reloaded at least once)."
"Grenade cartridge -- without visible bullet, lengthened 61 mm lacquered steel case with conical neck, yellow, red, or black primer annulus according to type of grenade, short, hollow, conical wooden bullet acting as closure."
"Grenade cartridge -- short, rounded-nose wooden bullet, lacquered steel case."
etc. etc. etc.
Germany was also not the only nation to use blanks or grenade launching cartridges made in such a fashion.
France (8mm Lebel and 7.5x54), Japan (7.7), Britain (.303 for use in the Bren gun), Finland (7.62x39), and even the United States used wooden bullets...
For the .30-06...
"Cartridge, Rifle Grenade, for Viven-Bessieres training grenade, natural wooden bullet and "V.B." in head stamp, 0.32 gr. load of black powder and 3.26 gr. of smokeless powder."
But, for the most part, in the United States blanks and rifle grenade cartridges either had hard paper "bullets" or were simply closed with an overwad covered with lacquer.
I have several different European and American cartridges of this type in my collection.