Unfortunately, a sponge mop or Swiffer will move the dirt and grit around and get trapped in the grout lines. Use only a string mop and change any dirty or even cloudy water. If the final wash isn't with clear water, you'll get hazing on ceramic tile. A swiffer is probably OK for laminate...

I use water and vinegar for my tile and laminate. I use a mop with a removeable microfiber cloth. I take it off, soak it, wring it extremely well, mop and repeat when needed. I vacuum first before mopping (my vacuum has a bare floor setting). I think I got the mop at Lowes or Bed Bath Beyond. Main thing with laminate - do not spray anything directly on it and make sure any mop you use is wrung out very well. You do not want to saturate the floor.

Microfiber. The old ways/products are good, but innovation and technology makes them better.

Here is a link that might be useful: Microfiber Swivel Household Mop

Steam mop will work, you dont want to regular mop laminate.