The Fibonacci cube $Γ_n$ is the subgraph of the hypercube $Q_n$ induced by vertices with no consecutive $1$s. Munarini introduced Pell graphs, a variation of Fibonacci cubes defined on ternary strings. A generalization of Pell graphs to $(k+1)$-ary strings has recently been proposed. In this paper we introduce Munarini graphs, which constitute an alternative generalization of Fibonacci cubes and Pell graphs. One of the main advantages of Munarini graphs is that, unlike previously proposed generalization, they are daisy cubes, as are Fibonacci cubes and Pell graphs. In this first article, we study some of their fundamental properties including the size, the recursive structure, the cube and maximal cube polynomials.