Depends on what you'd consider a problem. But they were essentially the same as what existed before, with the exception of the things Caesar had fixed or ameliorated during his dictatorship (fixing the calendar, founding colonies to provide new living space for citizens who might have struggled to make a living in Italy, etc). For instance, the senatorial class was obscenely wealthy compared to the working class, and social mobility was very limited. The super-rich controlled a huge proportion of the land in Italy, and one of the most contentious issues of the late Republic was the suffering of small landowners, who were theoretically supposed to be the lifeblood of Rome. In the provinces, Roman governors were all but expected to milk the locals for riches on a significant scale; as long as this didn't harm Roman interests or influential people, no one was going to stop them. There was also very little check on the power of these governors, other than removal by the Senate or prosecution after their term expired, leading me to the next point: bribery of juries was rampant. With sufficient money, almost anyone could be acquitted, and as a result the issue of who could sit on a jury to try a senator was a thorny problem in the late Republic. Rome itself contained masses of very poor people subsisting on state-subsidized grain, a supply which after Caesar's death was under threat: much of Rome's grain came from Sicily at the time, but Sicily had fallen under the control of the rebel Sextus Pompeius, who would build up a powerful fleet with which he wielded disproportionate power due to his ability to threaten the grain supply. And essential precedents had been set in the preceding decades for politicized violence in Rome, as well as for political strongmen to use military force for their own ends.

EDIT: Also, bribery was common in elections to the various offices of the Republic, in addition to the juries I mentioned. Partly because of this, and also because of factors such as the disproportionate clout wielded by a few aristocratic families (although in fairness, after Caesar's dictatorship, these families were in much weaker positions), those elections were by no means meritocratic. Meaning that not only was there little check on the rapacity of Roman governors, the mechanisms intended to ensure that they were at least men of good quality were fairly flawed.