State-of-the-art low-dose CT methods reduce the x-ray tube current and use iterative reconstruction methods to denoise the resulting images. However, due to compromises between denoising and image quality, only moderate dose reductions up to 30-40% are accepted in clinical practice. An alternative approach is to reduce the number of x-ray projections and use compressed sensing to reconstruct the full-tube-current undersampled data. This idea was recognized in the early days of compressed sensing and proposals for CT dose reduction appeared soon afterwards. However, no practical means of undersampling has yet been demonstrated in the challenging environment of a rapidly rotating CT gantry. In this work, we propose a moving multislit collimator as a practical incoherent undersampling scheme for compressed sensing CT and evaluate its application for radiation dose reduction. The proposed collimator is composed of narrow slits and moves linearly along the slice dimension (z), to interrupt the incident beam in different slices for each x-ray tube angle (θ). The reduced projection dataset is then reconstructed using a sparse approach, where 3D image gradients are employed to enforce sparsity. The effects of the collimator slits on the beam profile were measured and represented as a continuous slice profile. SparseCT was tested using retrospective undersampling and compared against commercial current-reduction techniques on phantoms and in vivo studies. Initial results suggest that SparseCT may enable higher performance than current-reduction, particularly for high dose reduction factors.