Episode 9: Beatbox Bandits/Evil Spirits:

That jogging guy: Not only would this cheerful fella almost certainly not have been running just to get exercise, a concept that didn't exist yet, but he would definitely not refer to it as "jogging", a term first applied to athletic running in the late 1960's & 1970's. Though running as a competitive sport goes back as far as the concept of sport itself, it wasn't introduced in Japan until after the Meiji Restoration (1868) along with other Western sports such as baseball. (Up until then, Japanese competitive sports were all derived from warrior skills: archery, kendo, jujitsu and the like.)

Another example of Japlish: The checkpoint guard says "Bingo!" when he finds the body he's after in the hut.

Not really an anachronism, but: The revolutionary rhetoric and the importance of marijuana to the plot are certainly intended to make one think of the general climate of America's 1960s, though there are no direct points of comparison I can see. However, some have pointed out that the souhei master's inflammatory speech is reminiscent of crowd addresses by the likes of Malcolm X and Che Guevara.

"The material that erases all hate": Marijuana has been used as an agent for achieving euphoria since ancient times; it was described in a Chinese medical compendium traditionally considered to date from 2737 BC. Its use spread from China to India and then to N Africa and reached Europe at least as early as AD 500. A major crop in colonial North America, marijuana (hemp) was grown as a source of fiber. It was extensively cultivated during World War II, when Asian sources of hemp were cut off.

Anachronisms Occurring Only in the English Dub:

Erik Alan points out that Old Yamane in his closing monologue refers to "that Summer of Love", a phrase commonly used in America to refer to the summer of 1967, the full bloom of the hippie era. As such it's almost certainly intended to connect with the general reference to the 1960s noted above.

--[Just to have it said, this is not what the phrase connotes in the anime Eureka seveN.]