Author's Saving Throw: The Emperor Edition's changes to the gameplay, and also fixing a large number of the bugs that had plagued the game since release.
Demonic Spiders: Oathsworn. A very powerful sword infantry unit of the barbarian tribes that only other late-game infantry might stand a remote chance against head-on and makes any artillery you have worth its salt just for withering them down a bit beforehand. The thing is though, Oathsworn are a general's bodyguard unit right from the start of the game, meaning you might end up seeing at least one in any given barbarian army. Many players have found a desperate or early battle against barbarians ending in tears as the Oathsworn tear apart their troops, or shock as the final general's bodyguard unit pulls off a very good Taking You with Me moment and heavily damages all comers despite being surrounded on all sides. In the absence of the aforementioned counter units, trying to hold them off with hoplites/pikemen/toughest infantry unit you have before filling their unshielded end with javelins or sling bullets/cycle-charging them with shock cavalry is highly advised to not see "Losing combat decisively" over all of your units fighting with the Oathsworn.
Hype Backlash: While reviews have for the most part been favorable (currently ranking 76% on Metacritic), the sheer anticipation and buildup leading up to the game's release have undoubtedly caused this among some, with criticisms aimed mainly at the AI and pacing issues. That said, the developers and fandom at large seemed to have handled this well, with the notable exception of TWCenter. It's reached the point that Creative Assembly actually apologized for how the game fell short of both the developers' and fans' expectations.
It's Easy, So It Sucks: Longtime fans have complained that the game is "dumbed down for the casuals" due to things like the flag capture mode and the lack of a family tree.
Loads and Loads of Loading: One of the largest complaints about the game is due to the rather long time it takes when cycling through a turn. This is probably due in large part to the Loads and Loads of Factions that exist in this game, far in excess of any previous Total War game. Patch 2 speeds this up quite a bit, especially early on.
Love It or Hate It: Depending on who you ask, the game is either a nicely done if flawed entry into the Total War series or a horrible bug-ridden letdown, with completely no middle ground allowed. It's quite important when you catch up with the game - right after the release, it really was a horrible bug-ridden letdown. Patches fixed most of the issues, so if you start playing after their release, the game is much improved. It also depends on the forums: mention it on TWCenter and you got an 10-page Flame Warat minimum.
Silanus, who rises to Proconsul during the course of the Prologue campaign. After conquering the Samnite capital, and with one of Rome's two consuls already dead at Capua, Silanus then kills the remaining consul in order to pave the way for his own rise to power. Maybe.
Arminius, the antagonist in the Teutoburg Forest historical battle and the one AI commander you cannot flatout defeat in the game (even if that's partially from not playing fair on a meta level).
If it helps, in Real Life after Teutoburg Forest the Romans completely lost their shit and sent expedition after expedition after Arminius, who spent the rest of his days running for his life. He finally died at the hands of his own tribesmen who saw him as more trouble than he was worth.
"The men are wavering." Not to mention that it's always preceded by said unit saying something like "By your command!" or "Forever loyal!" before running away like a bunch of silly ninnies.
The constant grunting and shouting of Champions.
The blaring sound that accompanies alert messages. After a string of bad events (enemy agent activity, cities taken, generals dead, ect) you could be inundated by alert messages, each of which playing that loud grating noise. It can serve to compound any frustration you might have been feeling from the bad events that prompted them.
Rescued from the Scrappy Heap: The Realm Divide Scripted Event introduced in Shogun 2 as the endgame event has been tweaked into a much less penalizing Civil War mechanic. Gain enough political power in your faction, and the other political factions will rebel against you, instead of having every other faction in the game become hostile.
Right after release, the flag-capturing mechanic in field battles proved to be incredibly unpopular, with many players claiming it ruined the tactical depth of battles.
The aforementioned Civil War has its detractors, as multiple fully sized advanced armies are spawned in wherever the Civil War begins, usually your capital. Depending on how late in the campaign it happens, this can go up to 15 armies and fleets ready to ravage your core provinces. It doesn't help that the chance of happening is completely random.
With the changes in the Emperor Edition, now the dislike is generally from the fact that the defecting units are drawn from your existing armies and fleets, accompanying agents and the regions they're stationed in, which can be painful when a prized 10-star army and/or general leaves and all their traditions and skills are lost to you.
The politics mechanics by proxy are disliked given how a player has to outright intend to increase their political power for its bonuses at the cost of causing the risk of Civil War. To a lot of players, there's not really any point to getting very involved with it unless you happen to get unlucky enough that you're so unpopular it's causing penalties.
Transport ships, which at least before the patches, could easily beat real naval units.
The interface as a whole was this to some people, but what most people agree on is that the pottery-like unit cards are hard to distinguish from one another, unlike Shogun 2's which get the trifecta of being beautiful, thematic and distinctive.
Almost all non-bug related complaints about the game usually come down to how the detractors believe the first Rome did it better, ranging from major mechanics like the economy to minor things like faction colours or the way roads are handled.
The UI's change to how the tech tree works caused some fans to complain, as well as the "flag capture" mode alienating longtime fans.
The family tree has been replaced with a senate system, listing only the surviving adult males of your own faction (either the family you chose as Rome or Carthage, or the royal bloodline, or the Elder Chiefs for barbarian factions), and those of the other families. Although a family tree's not quite as relevant in Rome II as it was for Rome and Medieval II, the change is a major overhaul which takes a while to get used to. Plus, it was really useful to see how many potential generals or marriageable family members you had at your disposal, while the current system feels rather random.
Tough Act to Follow: Tied to Hype Backlash above. While the game clearly aimed for greatness, in its early state it fell short of what Total War: Shogun 2 (or for its fandom, the original Rome) achieved. It didn't help that the game was heavily hyped by Creative Assembly and being made for years with a massive budget, leading to over-the-top expectations from players. By now, patches and DLCs have managed to fix almost all of the technical issues (the design issues are a matter of debate) and greatly expanded the replay value, but the title's reputation has already been tarnished by its initial problems.