This is a very narrow answer but it's of relevance. Tacitus was a senator who wrote his own "Annals" during the 'Principate', sometime around 90-110 AD (actual unknown). He had access to the Senatorial archives so historians tend to think of his work as more representative than most. This was basically a history of Roman emperors (generalizing).

Now, Tacitus clearly longed for the Republic. Or at least an idealization of 'Republic'. He writes, after Augustus' death (in the history, not in reality), "a few voices began idly to discuss the blessings of freedom" in public circles. (Tacitus "Annals" Book 1.4.1) The 'blessings of freedom' are the blessings of Republican Rome. Tacitus also laments, "few indeed were left who had seen the Republic" after the passing of Augustus. (Annals 1.3.6)

So, there were Romans, even senators, who longed for Republic. That's as narrow of an answer that I want to give, and that I feel capable of giving. There's a lot of nuance here, as usual. Whether the idealization of Republic for which they longed ever really matched the situation in Rome is up for debate. Also "regression" is an iffy term to use here, as it opens a bunch of can of worms about "what is best" a.k.a. what is progression, regression, etc. which is not a goal of history/historians.

Sources:

Mellor, Ronald. Tacitus' Annals. Oxford New York: Oxford University Press, 2011. Print.

Norena, Carlos. UC Berkeley. Lecture on Imperial Rome, HISTORY2: Ancient Empires, September 4, 2014. Presentation/Lecture.

This is my first post, how is it?

Unfortunately we can't say what the Romans as a whole thought, but we can say what a few individual Romans thought, or at least form an approximation. The classic example is Tacitus, but his views were not nearly as simplistic as the current top poster presents. Tacitus' least known work, for example, the Dialogus presents a variety of different perspectives on the transition from Republic to Empire with no clear indication of how Tacitus expects us to feel. On one hand, he certainly admired the consequential nature of actions in the Republic, when great men did great deeds, but he was also very aware of the violence and brutality of the civil wars. He decries the slavishness of the Senate under Domitian in the Agricola, but this does not mean he is arguing for a return to the Republic. He is an extremely complicated author and when parsing his sentiments we must be extremely cautious not to allow ourselves to be driven entirely by a few pithy anti-tyrannical bon mots.

Beyond that, it seems that Romans in general considered the Late Republic to be the grandest period of Roman history, by which I mean the most significant and most interesting. The great majority of Plutarch's Romans, for example, lived in the century before Augustus and figures and scenes of the period pop up again and again in Latin literature. This is even true to this day--even in Latin classes it is pretty common to find people fairly ignorant of Roman history after the Republic, or at least after the Julio-Claudians. When HBO sets a TV show in Rome they choose the Late Republic, and when a book is written on it Cicero is the lead character. So the Late Republic was consequential, but this does not mean that it was longed for.

In a general sense we can say that the vast majority outside of Rome itself was much more comfortable under the Principate, which smoothed out many of the brutalities of the Republican imperial system, and this can be seen in the wild popularity of the Augustus cult. Even within the city itself there was no real attempt to dismantle the monarchical system and it is pretty likely that the populace as a whole wanted nothing to do with that. Republican nostalgia, to what extent it existed anywhere, was confined to a small group of elite and even for them it is fairly unlikely that many seriously desired it. The imperial system worked.