Taking into account the circular economy guidelines and results of life cycle analyses of various materials, it was proposed to use a blackcurrant pomace filler in the production process of viscoelastic polyurethane (PUR) foams intended for application as mattresses, pillows, or elements for orthopedics. Open-cell viscoelastic PUR foams containing 10–60 per hundred polyols (php) blackcurrant pomace were prepared. It was found that after introducing the filler to the PUR foam formulation, the speed of the first stage of the foaming process significantly decreases, the maximum temperature achieved during the synthesis drops (by 30 °C for the foam containing 40 php of filler compared to unfilled foam), and the maximum pressure achieved during the synthesis of foam containing 20 php is reduced by approximately 57% compared to the foam without filler. The growth time of the foams increases with increasing the amount of introduced filler; for the foam containing 60 php, the time is extended even by about 24%. The effect of the filler on the physical, morphological, mechanical, and functional performances of PUR foam composites has been analyzed. The use of 60 php as the filler reduced the hardness of the foams by approximately 30% and increased their comfort factor from 3 to 5.