Spring wheat was grown on a loess-derived Luvisol under the conditions of two farming systems (conventional and organic) and five forecrops (sugar beet, spring barley, red clover, winter wheat, and oat) over the period 2021–2023. In the conventional system, mineral NPK fertilization and pesticides (herbicides, fungicide, insecticide, and retardant) were applied at the recommended rates for wheat. Mechanical weed control was also used (double harrowing). In the organic system, the organic fertilizer Humac Agro was applied and the fields under the wheat were harrowed twice. No plant protection products were used under organic farming conditions. The organic system was proven to have an effect on reducing spring wheat yield, on average by 23%, compared to the conventional system (the grain yield was, respectively, 4.59 t ha−1 compared to 5.96 t ha−1). In spite of the lower yield potential, the organic cultivation of spring wheat significantly improved the quality and health-promoting parameters of this cereal grain. Except for the total nitrogen and potassium content, the organic system contributed to a significant increase in the grain content of total dietary fiber (by 0.89 p.p.), o-dihydroxyphenols (by about 19%), and polyphenols (by about 12%), and increased the content of the following elements: Se, Mg, Ca, Cu, Mn, Fe, and Zn. Among the forecrops, red clover and sugar beet had the most beneficial effect on grain quality (including the amino acid composition and EAAI index), followed by oat (especially under organic farming conditions). The other wheat forecrops (spring barley and winter wheat) clearly deteriorated the yield and quality of spring wheat grain. To sum up the obtained research results, appropriate management of organic spring wheat cultivation (forecrop sugar beet or red clover, Humac Agro fertilizer) contributes to high grain nutritional quality relative to the conventional system and also reduces the yield gap relative to conventional farming.