The Romans did so, but didn't set up the games as an annual affair. If one believes in the gods as the ones who send the plague, one could say that failure was why four years after the original games were held, a plague struck Rome. To help get rid of it, the Romans made the Ludi Apollinares permanent, with the praetor urbanus, P. Licinius Varus, fixing the date at 6 July, according to the Smith entry; elsewhere (e.g., St. Andrews: Ludi Romani: holidays and holy days under the Republic [AN3028L03Ludi.pdf]) it is listed as July 13. The Romans gradually expanded the Ludi Apollinares to eight days.
Roman games meant theatrical productions and/or arena entertainment. The Ludi Apollinares were for circus games (which did not include gladiatorial games during the period of the Republic) and theatrical productions. The Praetor Urbanus was in charge of the Ludi Apollinares. In addition to games, there were sacrifices in honor of Apollo and other deities. In Festivals and Ceremonies of the Roman Republic, H.H. Scullard specifies that the sacrifices were a gilded horned ox for Apollo, a gilded horned cow for his mother Latona, and gilded-horned she-goats for his sister Diana.
Lily Ross Taylor, a 20th century Classicist and the first female fellow of the American Academy in Rome, says the circus games were more expensive and important than the theatrical games, so the ludi circenses part of the Ludi Apollinares was reserved for the last day; the other days were devoted to ludi scaenici. In the early empire there were seven days of ludi scaenici preceding the one of ludi circenses.
"The Opportunities for Dramatic Performances in the Time of Plautus and Terence"
Lily Ross Taylor
Transactions and Proceedings of the American Philological Association Vol. 68, (1937), pp. 284-304