Fantastic question. (edit: here is a decent map to get one's bearings). This is not a case of the noble Volscians being able to pull off a technological feat which was beyond the Romans of later times. Instead, the region underwent a profound ecological shift in a rather short time which transformed a fertile, relatively pleasant agricultural region in the 4th century BCE into a deadly, unpleasant swamp by the middle of the 1st century BCE. This shift was almost certainly due in large part to human activity.

When the Volsci took over the area in the early 5th century BCE, it was an agricultural powerhouse. In fact, it was the go-to place for Romans to source grain when their own supplies were not sufficient. So in 508 BCE, the Romans sent agents to buy from the Volscians, and again in 492. Pliny the Elder, NH 3.5.59, says that there had been twenty-four cities in the greater Pontine region, all ruined and/or lost by his own time (1st century CE). It is clear that the area of the marsh had been much less "marshy" in times past, with much more usable land. But it isn't clear that construction by the Volscians made it so (see below), or if the landscape was just very different in those earlier times. Arguments that the area had always been marshy and uninhabited since prehistory are hard to defend. The Romans were very interested in the Pontine plain, and it is often identified as their first major imperialistic goal. They warred against the Volsci consistently in the 5th and 4th centuries BCE to this end. Wealthy Roman clans quickly moved to gobble up land holdings on the Pontine plain once it was acquired, and in 383 BCE a Board of Five was set up to divide it up (Livy 6.21.4). Two new Roman citizen tribes were created (Ufentina and Pomptina), and the southern Pontine region was named the ager Pomptinus, clearly an agricultural zone and not a dangerous marsh. It would not be called "marshes" (Pomptinae paludes) until the 1st century BCE. Italian archaeologists even claim to have detected markers of the Roman centuriation there (the dividing up of land into regular gridded plots; see maps here, figure 2). The 4th century was a time of intensive Roman cultivation in the Pontine plain, and, at the same time, the inevitable absorption/dislocation of the remaining Volsci. The massive Dutch regional survey of the Pontine plain shows that the area had been host to humans since the Neaderthals, had witnessed increased activity at the beginning of the Iron Age (probably in conjunction with the arrival of the Indo-Europeans), and that in the 4th century BCE, the sheer number of sites greatly increased. All of this jives with the historical evidence we have.

The canals mentioned in OP's question probably refer to the cuniculi, underground tunnels connected to the surface by vertical shafts, very similar to modern-day storm sewers. They are typically built of stone and they are virtually indestructible. There are many of them known, and probably many many more that lay undiscovered in places that have undergone significant landform change. They are scattered around central Italy and there is a lot of debate about their origins, dating, and purpose. Date estimates range from the 9th century BCE to the 300s BCE, and some are clearly much later than that, though not very different in form. It used to be thought that the technology was Etruscan, and indeed some of them are clearly linked with Etruscan roads, but others are clearly linked with Roman villas with no evidence of pre-Roman occupation. The whole subject is quite confusing and requires a great deal of specialization. In function and form they are very similar to the most famous of them all, the Cloaca Maxima, which drained the swampy low-lying areas of Rome among the seven hills. The largest are associated with the town of Veii, Rome's first major adversary. Both the Lago di Nemi and the Lago di Albano have large cuniculi drainage. There was an old prophecy that the Romans would never capture Veii "until the Alban Lake be drained" (Livy 5.15.2–16.1, 16.8–11).

The cuniculi in the "Pontine marsh" area all seem to date to the Roman period of intensive occupation, the 4th century BCE, and not earlier, which is problematic for attributing Volscian activity to their creation. I am not an expert on the cuniculi by any means, though. The area north of the Marsh called Velletri has cuniculi probably constructed in the 4th century BCE, during intensive Roman activity there, though some push that date back into the 6th century. It is an incredibly specific area of expertise and unfortunately not mine. OP, if you know of a recent study which pushes the date of the Pontine cuniculi backwards into the 5th or 6th century BCE, that would make them Volsci works rather than Roman. But even then, their construction was not limited to the Volsci, as the examples at Rome and Veii show.

After the Romans take over the Pontine region in the 4th century, we unfortunately get a large gap in our sources, and the Dutch survey does not show anything but the big picture. We hear nothing of the Ager Pomptinus for two centuries. In 312 BCE, however, a major event impacts the region: Appius Claudius builds his famous road directly through it! The Via Appia ran from Rome to Capua and it goes right through the middle of the area which would later be called the Pontine Marshes. Archaeology shows that the road was built with consideration to the cuniculi there, and that the cuniculi must have been been functional at that time (I'm not sure how they arrive at that conclusion, though). The fact that the road runs through it also suggests that the Pontine plain had not become a swampy deathtrap by then (else why build a road right through it?). I agree with the vague theory that this is the key moment: the construction and presence of the Via Appia changed the Pontine plain in profound ways. We'd need a hydrologist-archaeologist hybrid to weigh in, really, but the physical transformations of the road seem to have seriously altered the movement of waters, and/or created standing, stagnant water where they had not been previously. We have a sense that malaria had probably been present in the Pontine region for many centuries (there was an ingens pestilentia "massive disease event" in 406 BCE which might have helped usher out the Volsci), but it was probably both worsening naturally through time and also was helped to worsen further by Roman intensive agriculture and then this massive road construction. By 36 BCE, Horace is noting that the stretch of the road through the Pontine region was mobbed with mosquitos (mali culices, Sat. 5.14-15). There had certainly been marshes in the Pontine plain before 312 (Livy says that the Volscian city of Anxur, later Terracina, had been surrounded by marshes in 406 BCE). Man-made activities like deforestation and excessive diversion of waterways probably contributed to the spread of the marshes, especially seasonal swamps which mosquitos prefer. The comprehensive Dutch survey of the Pontine area supports this rough timeline, with a massive population collapse sometime before the 1st century BCE and after the intensive Roman occupation in the 4th century BCE. The culprit of this disaster was broadly a profound change in the land, but the tip of the spear was malaria via Anopheles mosquitos.

Caesar was not the first to try to fix it, and would not be the last. In 160 BCE, the Roman Cornelius Cethegus attempted to rectify the marshy conditions of the Pontine, but was unsuccessful. The gradient of the region was too low to quickly carry away the waters via canals, even with Roman engineering expertise. The area was also probably already a veritable death sentence for the laborers. The efforts of Cethegus likely just displaced some of the water westward towards Ostia, intensifying malaria problems there and doing little or nothing to alleviate the problems in the Pontine marshes. The Romans were no strangers to massive waterworks projects, however. M. Curius Dentatus, in the early 3rd century BCE, had drained Lake Velinus without any apparent difficulty. The result was a broad, fertile plain where grass grew readily on ground that was still moist and muddy. The basin became famous for horse-rearing. It is likely that the Romans of Cethegus's time were well aware of the huge transformation of the Pontine plain, or were perhaps witnessing it within a few generations. They certainly had the capability to complete huge engineering projects, especially with water, but there is also a matter of political will, feasibility vs difficulty, cost, etc. So see here. It was likely the same for Caesar's intentions for the marshes, though he was assassinated before he had a chance to turn his attentions to the project fully.

Others after Caesar also set their eyes on the rehabilitation of the marshes, but not for many centuries. Pope Sixtus V (1585-90) sought via his bonifications to fix the swamps, but his efforts had largely the same effect as Cethegus's had. His ill-advised personal visit to the project probably led to his death via tertian fever (malaria). Napoleon could not stand the thought of such a famous and unpleasant place as the Pontine marshes within his empire, but, again, the question of political will, cost, risk vs reward, etc, seems to have taken its toll. In the late 18th century, Pope Pius VI gave it a go, again without success and again just making things worse for Ostia. Mussolini was the first to really effect a real remedial change on the marshes.