John,
I don't normaly get involved with hard surface cleaning, so this should not be considered as expert advice.
Some months ago whilst cleaning the carpets in a six-bed house, prior to my client movin in, I was asked to clean tiles similar to the ones you described, in a vwry large conservatory.
Bearing in mind that I had one equiped only for carpet cleaning, the method I used was:
1] Heavy pre-spray with solution of trafic lane cleaner plus neutral shampoo - both at half normal strength.
2] Agitated with Host machine with multi-directional passes. (The contra-rotating brushes working in the vertical plane 'lifting' the soil from the recesses, the shampoo foam helping to hold the soil)
3] Rinse with hard-floor wand on HWE, a weak acidic rinse agent (B109) in tank. Defoamer in waste tank.
By the end of stage 2 the floor was covered in a creamy grey sludge, and the end result most satisfactory.
Maybe the 'real pro' would snear at this method - all I can say is that it worked for this old carpet cleaner
,
Hope it helps you,
John.