Nowadays, effective and eco-friendly ways of using waste materials that could replace natural resources (for example, sand) in the production of concrete composites are highly sought. The article presents the results of research on geopolymer composites produced from two types of waste materials—hemp and fine fractions recovered from recycled cement concrete, which were both used as a replacement for standard sand. A total of two research experiments were conducted. In the first experiment, geopolymer mortars were made using the standard sand, which was substituted with recycled fines, from 0% to 30% by weight. In the second study, geopolymers containing organic filler were designed, where the variables were (i) the amount of hemp and the percent of sand by volume (0%, 2.5%, and 5%) and(ii) the amount of hydrated lime and the percent of fly ash (by weight) (0%, 2%, and 4%) that were prepared. In both cases, the basic properties of the prepared composites were determined, including their flexural strength, compressive strength, volume density in a dry and saturated state, and water absorption by weight. Observations of the microstructure of the geopolymers using an electron and optical microscope were also conducted. The test results show that both materials (hemp and recycled fines) and the appropriate selection of the proportions of mortar components and can produce composites with better physical and mechanical properties compared to mortars made of only natural sand. The detailed results show that recycled fines (RF) can be a valuable substitute for natural sand. The presence of 30% recycled fines (by weight) as a replacement for natural sand in the alkali-activated mortar increased its compressive strength by 26% and its flexural strength by 9% compared to control composites (compared to composites made entirely of sand without its alternatives). The good dispersion of both materials in the geopolymer matrix probably contributed to filling of the pores and reducing the water absorption of the composites. The use of hemp as a sand substitute generally caused a decrease in the strength properties of geopolymer mortar, but satisfactory results were achieved with the substitution of 2.5% hemp (by volume) as a replacement for standard sand (40 MPa for compressive strength, and 6.3MPa for flexural strength). Both of these waste materials could be used as a substitute for natural sand and are examples of an eco-friendly and sustainable substitution to save natural, non-renewable resources.