Raised k-Dyck paths are a generalization of k-Dyck paths that may both begin and end at nonzero height. In this paper, we develop closed formulas for the number of raised k-Dyck paths from (0,α) to (ℓ,β), for all height pairs α,β ≥ 0, all lengths ℓ ≥ 0, and all k ≥ 2. This represents a new approach to the enumeration of "simple paths with linear boundaries of rational slope", as discussed by Krattenthaler in his Handbook of Enumerative Combinatorics. We then expand upon Krattenthaler's results by enumerating raised k-Dyck paths with a fixed number of returns to ground, a fixed minimum height, and a fixed maximum height, presenting generating functions when closed formulas are not tractable. Specializing our results to either k = 2 or to α < k reveal further connections with preexisting results about height-bounded Dyck paths and "Dyck paths with a negative boundary", respectively.