After reading the following it seems that 3/4 inch would tend to cause the best players some trouble and that height was close to the minimum available until the '80s.
Excerpted from the USGA's website for instructional purposes:
Today's Fairway Conditions
The most typical fairway height of cut today is 1/2 inch. An appropriate management program combined with a mowing height of 1/2 inch produces a lie with a minimal amount of grass to be trapped between the ball and the clubface on a well-struck golf shot.
Mowing heights and management strategies at the other ends of the spectrum produce different types of lies and different levels of playability. For U.S. Open conditions, "We do not desire any grass between the clubface and the ball," reports USGA Director of Championship Agronomy Tim Moraghan. The absence of grass between the clubface and the ball offers the skilled player the best opportunity to control the spin and trajectory of the ball.
As fairway height increases above 1/2 inch, there is a greater likelihood that grass will be trapped between the ball and the clubface. The chance for "flier" lies is greater at taller mowing heights.
Mowing Technology
Mowing technology has improved significantly over the past 40 years. Old tractor-drawn 5- and 7-gang pull units of the 1960s and '70s were capable of mowing fairway turfgrass at just below 3/4 inch. Lower mowing heights were not possible with these machines because the bedknife would drag along the ground.
The next mowing advancement was the 9-gang self-contained unit, with 9-bladed reels that rotated based on the ground speed of the machine. The cutting units were ahead of the tractor for the first time ever. These machines cut the grass without bending the blades with the tractor tires. This produced a higher quality clip, but the mowing height still was limited to just below 3/4 of an inch.
In the early 1980s, another major advancement occurred with the development of the first hydraulically driven reels with either 5- or 7-gang reels drawn by a tractor. These mowers could cut the fairways at 1/2 inch, but the tractor was ahead of the reels, causing some bending of the grass blades due to the tires.
In the late 1980s, the first self-contained 5-plex units were developed. These mowers have cutting units ahead of the tractor, similar to the green triplex mowers. Today, these units are made with more durable materials to improve longevity. A grooved front roller has replaced the solid front roller, further reducing the bending of the turfgrass prior to mowing. Articulation of the cutting units also is highly advanced to reduce scalping injury to the turfgrass. Most 18-hole golf courses today have two 5-plex self-contained mowers that easily stay ahead of play with their speed. Mowing at 1/2 inch or even 3/8 inch is easily done with these 5-plex machines, and their cost is now more justifiable to practically all types of courses with either lease or purchase plans.
Turfgrass Species
The most popular turfgrass species used on golf courses today are hybrid bermudagrass, zoysiagrass, seashore paspalum, creeping bentgrass, Poa annua, perennial ryegrass, and Kentucky bluegrass. All these species except Kentucky bluegrass and zoysiagrass can be mowed well below 1/2 inch and can provide incredibly smooth and tight surfaces.