Well, this is what you've been waiting all semester for. Due to the popularity of Martian Successor Nadesico, the Gekiganger III OAV was created to service the fans. Unlike Nadesico, which is a parody of most sci-fi anime, Gekiganger III is not a parody at all; rather, it is more of an homage to the classic 70s robot shows that the creators watched long ago. I'm sure that this OAV was influenced by a near-countless number of anime, but it seems that most of the influence came from three specific series: Mazinger Z, Getta Robo, and Chodenchi Robo Combattler V.
Mazinger Z (1972) was the very first robot anime ever, and it was created by Go Nagai, who later created various series such as Getta Robo, Cutey Honey, Devilman, and more! If you look closely at Gekiganger III, you'll see an uncanny resemblance to Mazinger Z. Right down to the wings. Mazinger Z was built from "Super Alloy Z" to protect Earth. Coincidentally, Gekiganger was built from Gekiganium Alloy... to protect the Earth also. Several of Gekiganger's attacks are borrowed from Mazinger Z. Did you think Gekigan Punch and Gekigan Beam were actually original? They were taken from Mazinger's Rocket Punch and Laser Beam. Gekiganger V's Gekigan Fire was Mazinger's Breast Fire. For your information, Mazinger Z was brought over to the U.S. as Tranzor Z.
Getta Robo (1974) was the second robot anime ever and also was created by Go Nagai. Unlike Mazinger, Getta Robo was a combining robot. Three vehicles could combine in different combinations to form three unique giant robots: Getta-1, Getta-2, and Getta-3. Gekiganger III's three forms are Gekiganger III, Umiganger, and Rikuganger. Of course, some powers had to be borrowed from Getta Robo: Getta-2's Getta Vision and Getta-3's Getta Missile. Getta Robo has a Getta Beam; however, it does not shoot out from the eyes, like Mazinger or Gekiganger's beam attack. Gekiganger's Gekigan Spark and Gekigan Net were taken from Getta Robo G (1975). On an amusing note, even a side character was taken from Getta Robo. Jack King, pilot of Texas Mack, makes his way into Gekiganger III as Cowboy Johnny, pilot of Texas Robo. In case you were wondering, Getta Robo G made its way to the U.S. as Starvengers.
Chodenchi Robo Combattler V (1976) was a robot anime not done by Go Nagai. Combattler V was a robot that was made up by five vehicles to form this giant robot. Combattler V is a big influence on Gekiganger V. The mecha designs are very similar. They even have the same ultimate attack, Chodenchi (Gekigan) Spin. Gekiganger's Super Blazing Cracker is a heck of a lot like Combattler's Chodenchi Yo-Yo. And of course, the three pieces of Gekiganger, the Gekigan Jet, Gekigan Marine, and Gekigan Tank are taken straight out of Combattler: Battle Jet, Battle Marine, and Battle Tank. Missing are the Battle Crasher and Battle Craft. It is also worth noting that the character designs of Gekiganger III are very similar to those from Chodenchi Machine Voltes V (often called the sequel of Combattler V). Alas, Combattler V never made it to the U.S.
Obviously, this list is not extensive and I'm almost certain that more robot anime shows can be identified in Gekiganger III. I just pointed out what was obvious to me. My listing of Gekiganger's attacks is also far from complete. I'm still not even sure where Gekigan Flare came from. Shining Finger (G Gundam), maybe? Maybe not.