How to clean linoleum floors with texture?

If really grimy, you mist likely will have to use a scrubby with soap and water on your hands and knees. I do that for mother in law, it sucks.

Warm water, with baking soda and dish liquid should do the trick using a sponge mop.

I have vinyl textured floor tiles. Once a week I use a few squirts of Dawn dish liquid to one gallon hot water. Use a sponge mop. Once a month I use one cup (by eye) of lemon scented ammonia to one gallon hot water. Then I use Mop and Glo to give it a shine. The ammonia cuts through the dirt, grime, and built up wax.