cleaning a very dirty white tile floor

The kitchen floor in our new house is white press and stick tile (or I should say it was white at one point).  I want to rip it all out (and the adjoining dining room) and put in wood laminate, but that's not going to be able to happen for another year or two.  I'm just wondering if there's some miracle product that will get the years of grime out- there are a lot of gray patches on it from ground in dirt, it's kind of gross lol.

Is bleach the best thing?  Should I just let bleach soak on it or are there some other kind of cleaning products that are better?  Our previous house had darker brown tile, so stains were never a problem- I hardly used anything on them at all, so I don't know much about floor cleaning products.


Thanks so much!