We aimed to comparatively assess the prognostic preoperative value of the main peripheral blood components and their ratios—the systemic immune-inflammation index (SII), neutrophil-to-lymphocyte ratio (NLR), lymphocyte-to-monocyte ratio (LMR), and platelet-to-lymphocyte ratio (PLR)—to the use of artificial-neural-network analysis in determining undesired postoperative outcomes in colorectal cancer patients. Our retrospective study included 281 patients undergoing elective radical surgery for colorectal cancer in the last seven years. The preoperative values of SII, NLR, LMR, and PLR were analyzed in relation to postoperative complications, with a special emphasis on their ability to accurately predict the occurrence of anastomotic leak. A feed-forward fully connected multilayer perceptron network (MLP) was trained and tested alongside conventional statistical tools to assess the predictive value of the abovementioned blood markers in terms of sensitivity and specificity. Statistically significant differences and moderate correlation levels were observed for SII and NLR in predicting the anastomotic leak rate and degree of postoperative complications. No correlations were found between the LMR and PLR or the abovementioned outcomes. The MLP network analysis showed superior prediction value in terms of both sensitivity (0.78 ± 0.07; 0.74 ± 0.04; 0.71 ± 0.13) and specificity (0.81 ± 0.11; 0.69 ± 0.03; 0.9 ± 0.04) for all the given tasks. Preoperative SII and NLR appear to be modest prognostic factors for anastomotic leakage and overall morbidity. Using an artificial neural network offers superior prognostic results in the preoperative risk assessment for overall morbidity and anastomotic leak rate.