I would definitely not use a steam mop on a hardwood floor. I'm afraid there's nothing much safer than a damp mop or cloth, and lots of elbow grease.

Looks like the finish is mostly off of that wood which could be complicating things.

Lots of rags. Water + a snick of dawn soap (a DROP in a 1/2 gallon...and I mean 1 DROP of Dawn). Little bit at a time. Have a separate bucket for rinsing (water only). Rub dry when you are happy with the mud removal. This is probably hands and knees stuff.

Do NOT use a steamer. Ever.

And Sophie has spotted that these boards look very dry. That can be a good indicator that you have wear through (when raw wood is exposed). The raw wood will grab mud and never let it go (not until you do a full sand and refinish).

I would try to loosen the mud as much as possible (stiff bristle broom to PHYSICALLY scrape/remove the mud) and then vacuum. What is left is then soap/water/rinsed away.

Anytime you use soap (and something like this is a VERY special occasion...which is not a normal way to do this) you MUST rinse with water...possibly several times.

At this point (with what I can see) you probably have another 3-4 passes with "wet" stuff before you are done. And please look closer to see if you have raw wood exposed. It may be contributing to the headache...and would be an indicator that it is TIME to do a full sand and refinish.

First determine the kind of hardwood finish for choosing the cleaning method. Most of the new floors are polyurethane, urethane or ployacrylic which are water and stain resistant, thereby easy to clean. Older ones would be having shellac, lacquer or varnish finish. Saturate a rag mop in water and wring it till you feel it is slightly damp. First clear the floor, and dry mop or vacuum it. Try mop floor, going with the grain. If floors are polyurethaned dishwashing liquid and a dampened mop would do the job for you. Lastly, with a cloth buff floor. Check this article on tips for cleaning hardwood floors http://www.rbcclean.com/blog/cleaning-tips/tips-for-cleaning-victorian-hardwood-floors/ . In case of any doubt, choose gentler methods because hardwood floor isn't something you can take chance with. Else, seek guidance from high level industrial cleaning services. Old floors are easy ones to clean and not at all a pain ! Sure to vacuum initially to remove any loose soil or grit or even sweeping would do.

@Cancork Floor Inc. Thanks so much for the tips on getting mud out of hardwood floors. In my case it's clay. Your tip about using a stiff brush to get the clay out is going to help a lot!

Thanks!