"So the Ministry called upon Morfin. They did not need to question him, to use Veritaserum or Legilimency. He admitted to the murder on the spot, giving details only the murderer could know. He was proud, he said, to have killed the Muggles, had been awaiting his chance all these years. He handed over his wand, which was proved at once to have been used to kill the Riddles. And he permitted himself to be led off to Azkaban without a fight."
-- Dumbledore (HBP17)
Tom Riddle Jr. came looking for his maternal family in Little Hangleton and discovered a disheveled Morfin in the filthy Gaunt hovel. He learned of the history of his mother and father, then Stupefied Morfin. He went across to the Riddle house, murdered his father and grandparents with Morfin’s wand, then returned to the Gaunt house. There he placed a Memory Modification charm on Morfin so that when Ministry officials arrived, Morfin proudly boasted of killing the Muggles. Morfin’s wand proved to be the murder weapon and he was carted off to Azkaban.
The murders are said to have happened in the summer of Tom Riddle's sixteenth year.
"All he had to go upon was the single name 'Marvolo,' which he knew from those who ran the orphanage had been his mother's father's name. Finally, after painstaking research, through old books of Wizarding families, he discovered the existence of Slytherin's surviving line. In the summer of his sixteenth year, he left the orphanage to which he returned annually and set off to find his Gaunt relatives." (HBP17)
His sixteenth year, which would be the year he turned 16, was 1942.
We're guessing that he committed the murders in July, since summer break would be July and August. Why July? Because it doesn't seem like he would dawdle for a few weeks before tracking down his family -- he would be eager to learn more about them. Having the murders occur earlier in the summer would also give him plenty of time to perform whatever horrible rituals and spells are required for that murder to generate a Horcrux.
Interestingly, Harry wakes up with his scar hurting toward the end of August in 1994, so if the month and day of the week are correct for the murders, they would have taken place on August 23, 1944. But there's no reason to actually pin it down to that date when the year itself is off.