This is a slightly more morbid take, but the Germans were purported to have cut the fingers off of dead Senegalese soldiers in 1940 to obtain their rings.

This is allegedly because the Germans did not view the black soldiers as their equals, but as "armed savages, or beasts from the jungle". The battles between the French Senegalese soldiers and the Germans were particularly brutal, no quarter was given or asked. The black soldiers believed they would be murdered if taken prisoner, with some justification.

The take I took from this source is that Germans respected the bodies fellow soldiers that were of a similar races. Whereas races they viewed as "subhuman" (slavs & blacks), were the ones that were sent in for scientific analysis or had "trophies" removed.

There are few examples of trophy taking on the Eastern front by Nazi troops. However whole skeletons were shipped to Nazi scientists for racial studies. Not so much a trophy taking as a "studying" or "scientific" endeavor.

Source: Dark Trophies: Hunting and the Enemy Body in Modern War By Simon Harrison

EDIT:

I hope I didn't digress too much from the question.

I'm not sure about removing them from the corpses of fallen Americans, but one thing they had that would be a popuar find if Germans captured anywhere quickly evacuated would be cigarettes.

Prior to ww2 Germans (and now again in the present), like most Europeans, had developed a taste for the American tobaccos. War made many of the sources of this unwilling to trade with Germany or only via long, expensive dangerous routes, so the manufacturers in Germany turned to the more accessible but darker and stronger Asian and turkish blends, which lend themselves better to pipe smoking than cigarettes. (Saiz, Deutsche Soldaten, Uniforms, Equipement and personal effects of the german soldier 39-45 mentions this, along with some pictures of related artifacts, googl books).

So packets of Luckies, Camels and Marlboro would have been very welcome to German soldiers struggling by on roth händle, a brand that can cause serious smokers to cough up in surprise.

I don't have a lot of detail on this subject, but I do know that the Germans were quite fond of the M1 Thompson SMG and would frequently take them from American and British soldiers, either as a souvenir, or to use as a combat weapon due to its durability and high quality - particularly toward the end of the war, as Germany's war machine was falling apart and the quality of their own equipment began to deteriorate.

This was not a particularly common thing, however - the Thompson was chambered in .45, whereas most German arms were 9mm, so a lack of proper ammunition kept this from becoming a widespread practice, although it did happen some.