Technically, Route 31 is an east-west highway, though it's signed north-south in some places. Another bit of trivia: it's the only state highway (other than Route 80) to intersect both adjacent numbered routes (in this case, Route 30 and Route 32).
Route 31 was commissioned in 1942, heading from US 44 in Coventry to Route 32, for a length of 6.45 miles. The route followed Main Street (today's Route 31) for about 5.8 miles from US 44 to Depot Road; then it followed Depot Road easterly to Route 32, north of the current 31/32 intersection.
Five miles of Main Street leading from US 44 opened to traffic on Armistice Day, November 1923; a parade celebrated both events. Within a few years, this was designated as part of State Highway 315, which extended from today's US 44 to Depot Road at today's Route 32.
In 1932, SH 315 became part of US 6; the stretch of today's "Suicide 6" through Andover was part of a longer Route 87. In contrast to its unsafe reputation now, by 1938 the Andover route was considered a better, safer route than the old, rough, obsolete path of what is now Route 31. In 1940, US 6 was moved to the Andover route; the former US 6 through Coventry became a short-lived US 6A. In 1942, US 6A became Route 31.
In the 1920s and 1930s, the most hazardous part of "not yet Route 31" in Coventry was Depot Road: a curvy road with a narrow bridge and a railroad grade crossing. Officials in area towns asked the state in 1935 to move the highway off Depot Road to a more direct route to Willimantic.
In 1947, the state changed Route 31 in South Coventry to its present-day diagonal alignment. For a year, the Depot Road segment of former Route 31 was designated Route 31A, though I don't know if that was ever signed.
Today's Route 31 north of US 44 was originally not a state road. In 1951, the state added two roads:
In 1963, Route 31 was extended north along those two state roads to Route 74.