It has to do with the legend of the temple of Janus. The version of the legend I most recently heard was in a series of lectures on Roman emperors. I think. It may have been in a series of lectures on Greco-Roman civilization. Glancing around online (I’m at work, so I can’t spend much time) it appears my memory of the number of peaceful years was off, but not horribly:
In Rome, temples dedicated to Janus were numerous, the most important being known as the Ianus Geminus, a double-gated structure (one door facing the rising sun and the other, the setting sun) found on the Forum Romanum through which the Roman legionaries marched off to battle. This particular temple served a symbolic function. When the gates of the temple were closed, this represented peace within the Roman Empire. When the gates were open, it meant that Rome was at war. Between the reigns of Numa and Augustus, the gates were shut only once. Janus also had a temple on the Forum Olitorium and some time during the First Century, yet another temple was built in his honor on the Forum of Nerva. This particular temple had four portals known as the Ianus Quadrifons. At all temples, the priests of Janus sacrificed to him on a regular basis.
(found here: http://www.novareinna.com/festive/janus.html
and from http://penelope.uchicago.edu/~grout/encyclopaedia_romana/imperialfora/nerva/geminus.html
The doors of the Janus Geminus were opened to indicate that Rome was at war and closed during times of peace. Since the time of Numa, the doors were said to have been closed only in 235 BC, after the first Punic war; in 30 BC, after the battle of Actium; and several times during the reign of Augustus (for examples, when the Cantabrians were defeated in 25 BC, supposedly ending the Spanish wars; Livy, I.19; Suetonius, XXII; Plutarch, Numa, XX). Even in antiquity, the significance of the doors varied: whether peace was shut inside when the doors were closed or whether war was contained. In the Fasti, Ovid has peace released and war held captive behind barred doors (I.121-124) but then he locks peace inside (I.277-281). Or it may be that the god was thought to have gone out to assist Rome in her wars and stayed within the shrine to safeguard the city. Statius, too, has Janus withdraw behind his closed portal and then speaks of peace being put back there (Silvae, IV.1, 3). Dio recounts that, of all the honors extended to Octavian after the defeat of Antony and Cleopatra, “the action which pleased him more than all the decrees was the closing by the senate of the gates of Janus, implying that all their wars had entirely ceased” (Dio, LI.20.4).
Of course, that covers only 500 years of Roman history, not 1000, but since it was important for all of the emperors to appear as military victors, Rome did not get more peaceful after Augustus.
This is not a cite about 3 years, specifically; but if you’re looking for expert opinions on the character of Roman warfare, John Keegan wrote:
Roman imperialism was in large part the result of quite rational behaviour on the part of the Romans, but it also had dark and irrational roots. One of the most striking features of Roman warfare is its regularity – almost every year [for century after century] the Romans went out and did massive violence to someone – and this regularity gives the phenomenon a pathological character.
Here’s a page with more quotes about the pathological savagery of Roman warfare:
Random Internet discussion page on Roman warfare
Here’s a chilling contemporary quote from Roman historian Polybius on th sack of Carthage:
[Scipio Africanus] directed [his soldiers], according to the Roman custom, against the people in the city, telling them to kill everyone they met and to spare no one. The purpose of this custom is to strike terror. Accordingly one can see in cities captured by the Romans not only human beings who have been slaughtered, but even dogs sliced in two and the limbs of other animals cut off.
Admittedly the Carthaginians had pissed Rome off.
Wasn’t it just the Canal Zone, and not the whole country? As for the Phillipines, IIRC Roosevelt didn’t seize them, but Spain did hand them over as a consequence of losing the Spanish American war. Similar outcomes were common throughout the centuries when one colonial empire locked horrns with another, and things didn’t go so well. On the other hand what Roosevelt did do was harshly put down a local independence movement that sprang up around 1900