It's not that the Germans don't have a sense of humor; it is just that there are a lot of things they think they do not have permission to laugh at. Call it Teutonic Correctness (a variation of Political Correctness, or vice versa). This isn't just a German problem; it can be specific to different generations, too.
My grandparents on my father's side were born in the late 1800's, and were British ex-pats. Very different sense of humor from my parents; my parents saw the movie "Kind Hearts and Coronets", where the lead goes around killing all of the people that stand between himself and his inheriting an estate. It was a comedy, which my parents thought was hilarious, but which my grandparents were scandalized by. One could simply not laugh at murder! Definitely a generation gap: people from the late 19th century "Victorian" mindset most definitely had a long list of things Not To Be Laughed At. In a way, Monty Python can be seen as the ultimate breaking of this British Victorian taboo about what was, and what was not, funny.
Speaking of Germans, that very bad German, Adolf Hitler, actually had a pretty lively sense of humor, contra the dour and humorless image we have painted of him by war propaganda. He would often do impressions of important figures that he had met; apparently his impression of Mussolini had his audience in stitches. His mockery of FDR's intrusion into European affairs (where Hitler makes a long list of the countries FDR wants him to promise not to attack) is actually quite funny, if you are willing to surrender your prejudices about Hitler and FDR and see the humor in it. Often the Devil has the best lines.
Something the Germans, Scandinavians, and other germanic peoples seem to share is a kind of literal mindedness which makes for good humor. They can be so literal minded that they can make humor at their own expense, exposing their tendency not to see the nuances of language.
Case in point, someone told me two examples of the earliest recorded "viking" jokes, which apparently are very old indeed. I'll use the name "Sven" since I can't remember the actual names used.
One joke tells the story of a man we will call Sven, who was a famous viking halberd fighter; he was the most dangerous when weilding his favorite weapon, the halberd (unclear on what viking halberds looked like; possibly a kind of spear with an extra long spearhead for slashing as well as for thrusting). Anyway, this particular viking is attacked by his enemies, who chase him, but lose sight of him until they reach a thatched house. Surmising that Sven is hiding in the house, they send out several people to scout around the outside of the house, and on the thatched roof. Like a flash, the halberd stabs out a window, and kills one of them. Then it stabs through the thatched roof, and kills another one of them. The survivors flee back to their leader, who asks "is Sven there?" One of them responds "I don't know, but his halberd certainly is!"
Shorter joke: two vikings are walking along, and they come to a crossroad, in the middle of which is a severed head. "Doesn't that look like Sven to you?" one of them asks the other. "No; Sven was taller" the other replies.
These are hilariously funny, if you keep in mind that these are Scandinavian "ya, sure, you betcha" types, who are apt to understand what you say in the most literal sense of the words, rather than in the usual way implied by the context.
And the ultimate punchline was that all those countries listed by Hitler that he promised not to attack.....he attacked.