Not the most attractive, but they do have heated floor mats that go on top of the floor.

Example: http://www.floormat.com/heated-mats/greener-heat-anti-fatigue-mats.html

Other options you might explore are heat lamps, space heaters, oil filled radiator, towel warmer, wall heater, etc...

Here is a link that might be useful: Floor Mat

How about a wall heater near the floor, they have them that install under the vanity and blow the warmth on to the floor. In the winter you could put a runner rug the length of the bathroom and that would surely help. I've dealt with the marble for 20 years and I'm doing the bathroom over and installing wood. I have wood throughout my home and it's always warm and softer if you fall. Getting older makes you take every precaution.

I think the electric baseboard idea has a secondary benefit of drying/heating those towels overhead!

If you are trying to heat a surface and not the air, you should look into infrared heaters.

"Some kind of inadequate leaky structure is immediately underneath. "

Not necessarily. Marble just feels cold. 70 degree marble feels like ice compared to 70 degree wood.

I would look at the insulation/air infiltration too. The problem is not necessarily near the floor. There were major infiltration and insulation problems in the attic over the bathroom that were making my floor feel so cold.

Marble is such a heat sink though, even if you have it at 70deg it will suck the heat out of your 90+deg feet. Fluffy rugs and slippers might be your solution.

You could look into electric floor mats that can be plugged in or hardwired (Rugbuddy is one). You would place a nice rug over top so you don't see the mat. I don't have one, so I can't comment on their effectiveness.

Wow. Thanks everyone for your suggestions.

Bishop8 - it's a fire hazard to have towels too close to a baseboard heater. I believe the distance must be 8" between the top of the heater and the bottom of the towels, which should be fine, except for people and kids who fling the towel over the bar so that it slides down almost to the floor.

Billl - yes, that's exactly what I want to do, heat the floor surface not so much the air. You said infrared heater. I think of those when I think of the old-fashioned bathroom ceiling heaters you still see occasionally in motels. Are you saying they make infrared baseboard or wall heaters? Do you know any brands you could point me to?

Davidro - as I mentioned in my OP, I am not going to pull up floor tile, because I would never be able to re-do the marble floor tile to match the existing marble tub surround and shower walls. Also, this bathroom is an interior ensuite to the master bedroom - i.e. they share the floor structure, and there are no outside walls in the bathroom. But the bedroom floor feels fine, because it is not marble. It is the marble that is the issue, not the floor structure underneath the marble.

Birdgardner - I thought of the attic too, which is actually little more than a crawl space above the bathroom. I checked up there again, and it is quite well-insulated. As I mentioned in my OP, you can feel the marble through slippers, and I should have mentioned they are the thick slippers which are normally warm!

I am thinking of chucking the whole idea of a heater (and the cost) and getting a thick rug. Maybe I'll post a new thread on this Forum to ask for suggestions for bathroom rugs. Only problems with a rug is that it won't cover the entire standing area, and I am not a fan of rugs in bathrooms because they are dust/hair/powder/sprays traps. I guess there is no ideal solution, but I must weigh the pros and cons of all the solutions provided here. Thanks so much!

nothing i wrote was about ripping up any flooring,
neither to diagnose nor to fix the problem.
from the underside you go look and report back.
what is underneath this marble floor?
is it on a second floor? is it on a slab?
etc.
you must know; it's your house.

Davidro - the way you worded your post, asking what is "holding up the tile" and what is "immediately underneath" led me to the conclusion you wanted me to lift the tile. And your suggestion that I may need more insulation means I would have to open up the ceiling of the floor beneath. In answer to your question, the master bedroom and bath are on the third floor. Directly underneath those rooms is the 2nd floor dining room/living room. The ceiling of the 2nd floor is drywall. Although you say "you must know; it's your house", (I did not think the tone appropriate; I am looking for help here, not a setdown), it would be impossible to know what is between the two floors other than cutting a hole in the drywall ceiling. Luckily, I re-did the lighting on the 2nd floor recently, and the ceiling was open briefly. There are floor joists, there is insulation. You didn't specify what you are looking for when you directed that "from the underside you go look and report back." I do not want to open the drywall ceiling again, especially given that the bedroom floor adjacent to the bathroom is comfortable. As others have also mentioned, the problem is the marble tile; I just am requesting suggestions for ways to make the bathroom marble floor more comfortable. I appreciate the helpful suggestions on this thread that I will explore.

we have a marble floor in our mbr and i've put down some wool (relatively inexpensive) rugs... i love them!

given our floors are marble, i'm pretty careful about not spraying tons of products so there isn't alot of excess falling on the floors. we don't use spray deodorant, i don't use spray air fresheners in there, use very minimal hair spray, and generally spray perfume in either my closet or bedroom. the vacuum cleans the rugs well!! in fact i prefer vacuuming occasionally rather than getting on my hands and knees and cleaning which is what i seem to do for the rest of the tile...

I have not personally purchased one, but if you google "infrared bathroom heater", you'll see a lot of options from ceiling mounted to cove molding to baseboards. The infrared heaters work by shooting out infrared light. When that light hits a solid object, it gets absorbed and creates heat.

I think someone else mentioned it above, but when I was looking at heat options for my bath, I was seriously considering toe-kick heaters. They are installed under the cabinet in the toe-kick and blow warm air at your feet and would heat the floor in front of the vanity at least. I found this thread with a post about retrofitting.

Here is a link that might be useful: toe-kick heaters

Since most of the marble is not near the outside (envelope) it will not lose heat at a high rate. For this reason I vote for billl's proposed heat delivery system. The more heat you can get the stone to absorb, the greater will be your comfort level.

All the other options do something other than to put heat into the stone.

About: ".... Some kind of inadequate leaky structure is immediately underneath. ...." the answer is that it's the envelope and that nothing can be done about it from the ceiling underneath it.

About: "... Marble just feels cold...." the answer is yes, of course.

A radiant panel unit that would probably help in this situation. The unit, when placed properly, will radiate warmth onto the floor and tub - it may require ceiling mount.