What cover crops were popular before inorganic fertilizer?

That is an interesting question, that actually has several aspects to it. First, according to several source documents, the simple answer (according to Schoepf 1783-1784 author of ‘American Husbandry’) when cover crops were used, they were typically native grasses and clover with the field being left fallow. Generally a crop rotation was used as had been done for 1000’s of years. Colonists did not generally utilize cover crops to their benefit for various reasons, the main being cost and availability of labor. before 1800 virtually every American farm was a subsistence farm, and as such there would generally not be sufficient funds for purchase of specialized cover crop seed, neither for the labor necessary to plant a fallow field. Thus, what was at hand and could grow naturally dominated. For reference (highlighting the lack of a home market which affected purchase decisions by farmers of the period): in 1800, Pittsburgh, PA had just over 4000 people, and had the largest inland town population in the country. This being said, most farmers would find rotation of crops more cost effective than cover cropping since land was so abundant, and there wasn’t sufficient manpower available to invest in soil quality (unlike Europe). In fact, there are many records of colonial governors repeatedly justifying labor expenses to the ‘Crown’ and the trading companies as it was as much as 50 times the cost of labor in England (this is outside of a few coastal farms that were near a port and able to sell something for export to other colonies or abroad). Note that England, where land was (and still is) extremely expensive (with no open land of any consequence available) and labor cheap (serfs and end indentured servants where unemployment was rampant), there was more impetus for maximizing the efficiency of the soil available for farms. the Colonies, on the other hand, had plentiful land and labor was expensive (as you might expect, the population/acre were vastly different). The invention of the cultivator had a significant secondary impact on soil health as it reduced labor cost (much more can be written on this). An excellent book on tomato culture written in 1919 also mentions clover as the primary cover crop, but identifies that the cultivator improved the success of the cover crop. Additionally, the author noted that one tomato farm identified the the best mix as wood ash, clover and commercial fertilizer mixed with ‘well rotted stable manure’, so we see an interesting transition from pre-commercial fertilizer to commercial fertilizer alone with a mix of the two being used successfully. Note that this would likely not have a negative impact on soil organic matter (well known by this time) while maximizing agronomic efficiency.

According to a 1913 USDA Bulletin the following were being used: Alfalfa, Red Clover, Cowpea, Rye and Broom Sedge

According to a 1918 USDA Illustrated Lecture on Green Manuring, these were being used: Red Clover, Crimson Clover, Sweet Clover, Soybeans & Cowpeas, Velvet Bean, Bur Clover, Vetch, Field Peas, Lupines, Horse Bean, Rape, White Mustard, Rye, Millet, Buckwheat, Mexican Clover (from the Coffee family). 

According to a 1902 Bulletin from the Hatch Experiment Station at the Massachusetts Agricultural College the following were being used: Rye, Oats, Barley, Peas, Soybeans, Cow Pea, Hairy Vetch, Red Clover, Crimson Clover

According to a 1925 Bulletin from the University of California College of Agriculture Berseem Clover (also known as Egyptian Clover) was showing promise for maintaining soils. 

I also have a USDA 1935 Bulletin on Cover Crops but can't quite put my fingers on it at the moment...