Sparse grid discretisations allow for a severe decrease in the number of degrees of freedom for high dimensional problems. Recently, the corresponding hyperbolic cross fast Fourier transform has been shown to exhibit numerical instabilities already for moderate problem sizes. In contrast to standard sparse grids as spatial discretisation, we propose the use of oversampled lattice rules known from multivariate numerical integration. This allows for the highly efficient and perfectly stable evaluation and reconstruction of trigonometric polynomials using only one ordinary FFT. Subsampling such lattices randomly, we expect a further decrease in the computational complexity while preserving its numerical stability.