The need to model geometrically complex systems with improved ease of use and fidelity and the desire to extend the Tools for Sensitivity and UNcertainty Analysis Methodology Implementation (TSUNAMI) analysis to advanced applications have motivated the development of a methodology for calculating sensitivity coefficients in continuous-energy (CE) Monte Carlo applications. The Contributon-Linked eigenvalue sensitivity/Uncertainty estimation via Track length importance CHaracterization (CLUTCH) and Iterated Fission Probability (IFP) eigenvalue sensitivity methods were recently implemented in the CE KENO framework of the SCALE code system to enable TSUNAMI-3D to perform eigenvalue sensitivity calculations using CE Monte Carlo methods. This paper provides a detailed description of the theory behind the CLUTCH method and describes in detail its implementation. This work also explores the improvements in eigenvalue sensitivity coefficient accuracy that can be gained through use of CE sensitivity methods and compares several sensitivity methods in terms of computational efficiency and memory requirements. The IFP and CLUTCH methods produced sensitivity coefficient estimates that matched, and in some cases exceeded, the accuracy of those produced using the multigroup TSUNAMI-3D approach. The CLUTCH method was found to calculate sensitivity coefficients with the highest degree of efficiency and the lowest computational memory footprint for the problems examined.