MPC groove is kind of an urban legend. The real story is your cuts/edits are out of sync because there was no visual editor on the famed MPC3000 nor the MPC60. Even after you quantized, the samples were still off because of the splices. You can replicate this in your DAW software using the delay feature, adding or subtracting milliseconds to samples to add groove.
For example in Ableton Live, you can do the following to your samples:
The pad-based Akai MPC samplers were all the rage in the early hip-hop days and are still popular today, as they let producers tap out their own grooves. You don’t need an MPC to play your own groove – you could use any MIDI controller, or perhaps beatbox and use hitpoint editing.