"Destroying everything to do with" Antony is more than a bit of a stretch, and I'm tempted to accuse whoever made that podcast of simply talking out of his ass. There's some truth in what is said, though, so we'll start with that. When Octavian took Alexandria, says Plutarch, he had all of Antony's statues in the city taken down (though Plutarch says he left Cleopatra's up!). Plutarch mentions in his life of Cicero that during Octavian's consulship with Cicero Minor in 30 the senate voted to remove all honors from Antony, forbid the use of the name "Marcus" by any member of the Antonian family, and take down Antony's statues in Rome (note that Plutarch does not say that Octavian did this--indeed, it is implied that it may have been Cicero's idea, to get revenge for the treatment of his father). There were a few more punishments as well--Antony's birthday was declared an ill-omened day, and Suetonius relates that the senate declared him an enemy of Rome when he decided to support Cleopatra (who had already been declared to be the same thing). If surviving monumental evidence is a good indication, it seems that after Actium Antony's name was omitted from Augustan monuments (i.e. ones that Augustus set up)--he was probably, for example, omitted from the list of consuls on the Augustan Arch (which does not survive in any great quantity).

All that said, thorough though the state's (it does not appear to all be Octavian's doing) damnation of him, it's not as though he was totally wiped from all state proceedings or the memory of Italians. Nor was it forbidden to mention him--all that was really erased was his work within the state. The treatment of Antony should not be treated as a damnatio memoriae, which I suspect is what the creator of the podcast must be confusing it with. Damnatio memoriae consisted of erasing all possible evidence of a person's existence, beginning from his birth, and included wiping out not only public works and images, but also all written records of a person's existence (wiping him from the roll of citizens, eliminating him from all state records, etc). That's not what happened with Antony at all--he was dishonored, but not erased. Contemporary writers continued to write about him--Antony is a major figure, possibly the most prominent after Augustus himself, in Augustan literature, where he is characterized sometimes as a maniac, sometimes as hopelessly misled, and often as tragically doomed. Compare this to Cleopatra's treatment in Augustan literature-Cleopatra is never referred to by name, always as "the queen" or "that woman," and her character is almost uniformly damned (except for a few passages and a couple possibles--for example, if we take Dido or Camilla to be symbols for Cleopatra, which is not totally accepted). And Antony's family continued just fine--both Caligula and Claudius were direct descendants of Antony's. Further, ex-Antonians were all over the place (most of the Julio-Claudians were directly descended from them, as opposed to their adoption into the Caesarian family). It's been noted since Syme that it was not Antony with whom the war of Actium was fought, but Cleopatra, and Antony's disgrace was due to turning traitor, supporting Cleopatra after war had been declared against her.

Augustus also began to slowly rehabilitate Antony's image later in his life--Antony appears as one of the triumphators on Augustus' monument honoring the return of the Parthian standards (a major honor, especially considering that Antony had let his army starve to death in Armenia while campaigning against the Parthians). Already in Augustan literature we find a great deal of tragic sympathy for Antony, and certainly by Plutarch's day it was common to speak of him as some sort of tragically doomed figure. Antony was a very common literary character throughout the Principate and appears quite often. Whether these characterizations of Antony are at all accurate sort of depends on who's writing, but nevertheless we know that he was remembered and written about--Augustus didn't totally nuke his memory out of existence, nor had that been his intention. Besides, Cicero's most damning speeches, the invective Philippics, are aimed at Antony, and provide an enormous amount of historical material on him

EDIT: Besides, even if Antony's treatment had been harsher and he had actually been slapped with damnatio memoriae (a treatment which doesn't even exist until the Principate--as I recall, Sejanus was the first to have his name erased under damnatio memoriae), we would still know about him, almost certainly. Damnatio memoriae was never totally successful, from what we can tell, at erasing all evidence of a person's life, and in any case there was no legal ban on speaking or writing of the person. All three emperors to have suffered legal damnatio memoriae (Domitian, Geta, and Maximian) are known perfectly well, and often emperors that had been condemned were rehabilitated later (Commodus, for example). I should also mention that only the images of Antony in Rome and Alexandria were destroyed and that these were presumably public images. Private statues and masks almost certainly survived (I mean, we have statues of Antony still, so what does that tell you), and public images of him elsewhere probably did as well, as did inscriptions. Antony's coinage was not destroyed, despite the fact that it had his name on it and usually his face, and we have quite a lot of it. All this is just to reinforce the fact that "destroying everything to do with Antony" is hardly accurate--Octavian and the state set out to condemn his state work and dishonor his public name, not destroy his existence entirely