How did the Romans handle "fatbergs" in their sewage system?

The Romans did have problems with blocked sewers but much of the detritus that makes up the fatberg is modern - wet wipes, sanitary napkins, cotton buds and so on. Also, there are some other points to consider here, namely:

1. The main Roman sewer, the Cloaca Maxima, was originally built to drain land between the seven hills. It's primary function was not to act as a sewer in the modern sense, though that did become one of its uses.
2. Very few Roman residences, and not all public latrines, were connected to a sewer, even though they were legally allowed to do so. Primary sources do not say exactly why, but modern historians have surmised that (1) digging up roads to lay the connecting pipes would have been prohibitively expensive for most people, and (2) the Romans did not know how to put traps on their drains; consequently, there was a danger of odours and dangerous gasses (= explosions) coming back up the pipes, not to mention rodents and dirty water when the Tiber flooded.
3. Most Roman homes had cesspits where both human and kitchen waste was disposed of, or else waste could be dumped outside the city or (in the case of excrement) collected and used by farmers (see Where were ancient Romans supposed to dispose of their waste and garbage? for more details).
4. There was a flushing system which helped to clear the sewers, and there was also overspill from fountains and reservoirs. This is noted by Pliny the Elder (in Book XXXVI, The Natural History of Stones)

...there are seven rivers, made, by artificial channels, to flow beneath the city. Rushing onward, like so many impetuous torrents, they are compelled to carry off and sweep away all the sewerage

Also, the Roman engineer Frontinus (circa. 40 to 103 AD), referring to the supply of water in Rome, wrote in De Aquis

...there must necessarily be some overflow from the delivery tanks, this being proper not only for the health of our city, but also for use in the flushing of the sewers.

However, there was still a lot of waste that entered the sewers because it was not uncommon for people to simply dispose of waste on the streets, and both literary and archaeological evidence shows that the sewer tunnels needed to be unblocked from time to time as the flushing 'system' was not enough. For example,

a senator of the second century B.C., who wrote a history of Rome in Greek, ...[observed]...that “when the sewers had been neglected and were no longer passable for the water, the censors let out the cleaning and repairing of them at a thousand talents.”

Also, in 33 BC, (from Cassius Dio, Book 49, section 43):

Agrippa agreed to be made aedile and without taking anything from the public treasury repaired all the public buildings and all the roads, cleaned out the sewers, and sailed through them underground into the Tiber.

Silt was one problem, but other debris also blocked sewers. Most this came from waste that was just thrown on the street, both by residents and commercial establishments. This could include both

nonorganic (broken storage jars, lamps, bronze vessels, coins, and jewelry) and organic material (fishbones, eggshells, pips and seeds, and fragments of different kinds of cloth)

Literary sources also note that dead people were sometimes thrown into sewers: Cicero mentions this in For Sestius, as do Suetonius in his Life of Nero and Dionysius of Halicarnassus in Roman Antiquities, Book X, section 53 (corpses of plague victims).

We don't know exactly how the unblocking was done (presumably they used iron tools, and it was backbreaking work), but we do know that there were trained slaves (who also carried out repairs) and, from correspondence between Pliny the Younger and Trajan, convicts were also used.

For removal of the debris form the tunnels, Pliny the Elder mentions that the Cloaca Maxima was

of dimensions sufficiently large to admit of a waggon laden with hay passing along them

so quite possibly the slaves / convicts used a cart to help transport the silt and other detritus out of the sewer, perhaps together with a pulley system or crane where there were surface drain holes.

The state distinguished between public sewers (such as the Cloaca Maxima) and smaller, private ones. According to the jurist Ulpian (circa. 170 to 223-228 AD) upkeep of the former was the responsibility of the state, while the latter had to be maintained by the property owners.

Aside from the above, though, the

the literary evidence concerning...sewers in the Roman world is extremely meagre.

Other sources:

Ann Olga Koloski-Ostrow, 'Talking heads: what toilets and sewers tell us about ancient Roman sanitation' (2015)

Gregory S. Aldrete, 'Daily Life in the Roman City_ Rome, Pompeii, and Ostia' (2004)