We don’t know for sure. The Cloaca Maxima, the first great sewer, has traditionally been credited to the king Tarquinius Priscus. He was an Etruscan, and for a long time it was fashionable to credit much of early Rome’s early achievements such as public works to the Etruscans. However, it isn’t that simple:

In Rome the urban development of the low-lying valleys between the hills, and particularly of the Forum valley, required the reclamation of marshy areas by means of drainage. The achievement of this task is usually attributed to the Etruscans, who are thought to have been especially skilled in hydrology. The evidence is that the countryside of southern Etruria is riddled with drainage tunnels (cuniculi) designed to divert streams and to channel surface moisture away from waterlogged areas. Some of them are remarkable feats of hydraulic engineering. The trouble is that cuniculi of the same type can be found in Latium as well as south Etruria, and since they are impossible to date there is no way of telling whether the Etruscan ones are earlier or later than the Latin ones.

T.J. Cornell, the Beginnings of Rome

We do know that foreign engineers were brought in to work on various major public works. Later Roman architects freely acknowledge the debt they owe to others. Vitruvius, in his Architecture, is at pains to explain where various architectural methods or styles originated, or, at least, where he thought they originated.

Yet it seems that the Romans eventually took hydraulic engineering to levels much beyond anything any neighbours had managed. They were quite proud of what they accomplished. One water official wrote a book listing all the aqueducts and then exclaimed:

With such an array of indispensable structures carrying so many waters, compare, if you will, the idle Pyramids or the useless, though famous, works of the Greeks!

Sextus Julius Frontinus

They obviously believed these were a Roman invention, or at least something perfected by the Romans. A Greek writer, writing for a Greek audience, also says:

Indeed, in my opinion the three most magnificent works of Rome, in which the greatness of her empire is best seen, are the aqueducts, the paved roads and the construction of the sewers.

Dionysius of Halicarnassus

So while it is unclear exactly where the idea of such a system came from, the Romans definitely perfected it.