Ancient Greeks probably took mythology too seriously to record jokes about the gods. Most jokes regarding Greek gods seem fairly recent and are in a riddle format and easily understood by the general public and children. Whether or not intentional, some of the jokes are easily traced to stories in the Greek god's mythology.
Boom's Bardic Blog archives old jokes related to Greek gods that are primarily a play-on-words. It's not necessary to understand Greek mythology as the jokes have little to do with the referenced Greek god.
What kind of footwear does a Greek god wear? Tennis Zeus.
Which Greek god is the windiest? Ares.
How are ancient Greeks towels labelled? His And Hercules.
What is a favourite game of ancient Greek mythology? Hydra and go seek.
What is another name for Hercules's first labour? Finding Nemean.
How do you profess your love for a sea nymph? "I will love you whether you are Nereid or farid."
How did ancient microbes get around? With Scylla.
What is a game popular in ancient Greek mythology? Pick up Styx.
What sauce did Tantalus put on his food? Tartarus sauce.
This joke is another play-on-words, however, it relates to the mythology of the demigod Tantalus, whose father was Zeus. James Hunter for Encyclopedia Mythica notes that Tantalus was the king of Sipylos and invited to dine with the gods. Zeus had shared secrets at dinner, but Tantalus broke a trust by retelling them. His punishment was to forever be "tantalized" with hunger and thirst in Tartarus, a place of eternal torment in the underworld far below Hades. Fruits hung low, but when he reached out, winds would blow them away. Although Tantalus was up to his neck in water, he could never drink as the waters drained away as he lowered his head.
Where did the ancient Greek goddess have to go? To a Hera appointment.
It's not too far fetched to imagine that Hera, an Olympian god known for her beauty would have regular salon appointments. Greek culture adored physical beauty and Hera was the jealous wife of Zeus.
Why did the Satyr have such a head ache? He was hit on the head with a pan.
Greek-Gods.info, notes that the Satyrs were divine half-man half-goat creatures. Satyrs were the original party animals, following Dionysus the god of wine on his journeys, while drinking and dancing with the Nymphs. Pan, the most unattractive of all Satyrs was a god and the head of the Satyrs known for playing a flute. On that note: The word satire, for a comedic style of drama, can be traced back to the word Satyrs.