The original Rocky begins on November 25, 1975 and runs until January 1, 1976. Rocky II picks up directly after the title fight from Rocky (ie. January 1, 1976) and runs until November 25, 1976. On three separate occasions in Rocky III do characters make reference to "the last three years" (Paulie (Burt Young), Mick, and Rocky), which would seem to place the film in late 1979/early 1980. This is corroborated by Rocky's age in the film. In the original Rocky, he is 30 (meaning he was born in 1945). In Rocky II, he turned 31. If Rocky III takes place three years after Rocky II, Rocky should be 34, which is exactly what age he is (as pointed out on TV). All of this serves to illustrate that the film is set in 1979/1980.
It is not quite that simple however. Problems begin to arise in relation to Mick's death. His headstone records his date of death as August 15, 1981, placing the events in Rocky III almost five years after those in Rocky II (and thus making Rocky 36), leading many fans to accept this 1981 date as the 'correct' date. However, the headstone throws up further complications. In Rocky (set in 1975) Mick says that he is 76 years old, meaning he was born in either 1899 or 1900. If Rocky III occurs five years after Rocky II, and six years after Rocky, Mick should be 82 when he dies. However, his headstone lists his date of birth as 1905, meaning he was 76 when he died; the same age he was in a film set six years previously. To complicate things even further, the fight between Rocky and Thunderlips (Hulk Hogan), which takes place prior to Mick's death, is advertised as occurring in 1982! And if all of that wasn't bad enough, in Rocky V, Mick's death is stated to have occurred in 1982, not 1981.
In the end, fans are really left with one of two choices: the 1979 date (based on the "three years" comments and Rocky's age) or the 1981 date (based on Mick's headstone - dismissing the Thunderlips and Rocky V 1982 dates as goofs). The numerous inconsistencies on Mick's headstone would seem to undermine the validity of using the inscription to date the film, whereas the 1979 date is confirmed on four separate occasions in the film; once by Paulie, once by Rocky, once by Mick, and once by the TV newscaster when he says that Rocky is 34. As such, the most likely answer to the question of when the film is set is 1979/1980.
Rocky (1976): November 25, 1975 - January 1, 1976
Rocky II (1979): January 1, 1976 - November 25, 1976
Rocky III (1982): early-1979 - early-1980