Potato breeding must improve its efficiency by increasing the reliability of selection as well as identifying a promising germplasm for crossing. This study shows the prediction accuracy of genomic-estimated breeding values for several potato (Solanum tuberosum L.) breeding clones and the released cultivars that were evaluated at three locations in northern and southern Sweden for various traits. Three dosages of marker alleles [pseudo-diploid (A), additive tetrasomic polyploidy (B), and additive-non-additive tetrasomic polyploidy (C)] were considered in the genome-based prediction models, for single environments and multiple environments (accounting for the genotype-by-environment interaction or G × E), and for comparing two kernels, the conventional linear, Genomic Best Linear Unbiased Prediction (GBLUP) (GB), and the non-linear Gaussian kernel (GK), when used with the single-kernel genetic matrices of A, B, C, or when employing two-kernel genetic matrices in the model using the kernels from B and C for a single environment (models 1 and 2, respectively), and for multi-environments (models 3 and 4, respectively). Concerning the single site analyses, the trait with the highest prediction accuracy for all sites under A, B, C for model 1, model 2, and for GB and GK methods was tuber starch percentage. Another trait with relatively high prediction accuracy was the total tuber weight. Results show an increase in prediction accuracy of model 2 over model 1. Non-linear Gaussian kernel (GK) did not show any clear advantage over the linear kernel GBLUP (GB). Results from the multi-environments had prediction accuracy estimates (models 3 and 4) higher than those obtained from the single-environment analyses. Model 4 with GB was the best method in combination with the marker structure B for predicting most of the tuber traits. Most of the traits gave relatively high prediction accuracy under this combination of marker structure (A, B, C, and B-C), and methods GB and GK combined with the multi-environment with G × E model.