There were a couple of things that happened in the 70s. First there was the animated series that ran a couple of years in the mid 70's. But this was based on the original crew and the original cast provided the voices. Towards the end of the 70s, there was an attempt at reviving Star Trek. Tentatively called Star Trek: Phase II, it was to be the anchor of a new television network run by Paramount (The forerunner to the UPN network). William Shatner and Deforrest Kelley were set to reprise their roles, but no one else from the original cast was tentatively set to join the new tv series.
The plan for the new network was scrapped and work on Star Trek: Phase II was cancelled. They were well under way on the pre-production work, such as building sets and casting characters, but no episodes were ever shot and series died before it ever began. However, neither Rodenberry nor Paramount wanted to drop Star Trek entirely. So while Star Trek: Phase II, the tv series died, from it's ashes arose the first Star Trek movie. The original cast was brought back, along with one of the actors set to star in the Phase II series. This was Persis Khambatta, who played Lt. Illea.
So while yes, there was a plan for a second live series in the 70's, it died before it got off the ground and they switched and did movies instead.