What do you do when you're on-the-road and you cannot shower?

The obvious thing to do would be to just wash yourself with 2 hands or a sponge and soap/shower gel; or am I missing something here?

If you really can't handle the cold shower, why not just heat some water in a pot or kettle? Then mix it the with freezing water (if needed) and sponge-bath like the others suggest. I did this a couple of times when the hot water-supply was off.

Easy if you carry a portable heating-element.

If you have a thermos flask (admittedly unlikely) you could even heat the water before you go to to bed the previous night, so no time wasted in the morning. Works when the electricity or water supplies are unreliable, or there's a morning rush in the kitchen/bathroom.

In the shower, you stay aside and let some water in your open palms. Then sprinkle water over your body. Add more water gradually to let your body adapt. I think it's perilous to suddenly step into cold water.
You can do something similar over the sink.
I had to wash with cold water every day in the army and sometimes in student dormitories. I never liked it though others had no problem.

In some countries there are opportunities to shower or bathe that you might not expect.

• In Thailand, there are "police boxes" (?????????), which are mini police stations located on highways at the edge of towns. It seems more than half have showers you can use. Some have signs saying you're welcome to use them. At others just try asking the police. I find them always friendly and happy to meet travellers. Many are open 24 hours or have a shower accessible even when the police are not there. Some might only have a bucket shower. The water is never hot but it's a warm country so the water is never freezing either.
  In the very south, the police boxes are not great. When they don't have a sign in English you can avoid them. But instead they have even better facilities at "highway police" stations in similar locations. These are painted orange and brown and are larger and more modern. They have large bilingual signs advertising what they offer, sometimes even somewhere for cyclists to sleep!

• In Thailand and Cambodia you can ask at a Buddhist temple/monastery/pagoda if you can use their bathing area. Some only have bucket showers. I may be wrong but this is probably only appropriate for men. Personally I try to find a layperson to ask since I don't know the proper procedures to respectfully ask the monks directly.
  This might be possible in Laos and Myanmar too but I haven't tried.

• In Japan some internet cafes have showers. I assume some manga kissas and other similar places where people stay overnight in lieu of proper accommodation might also have showers. Some are free and some are coin operated.

• In South Korea you can always find a jjimjilbang (???) in any city or town. They will have saunas and hot baths as well. You must shower/bathe before using those other facilities. Note that once inside everybody will be naked but men and women will be in separate areas.

• In Japan, sent?s (??) seem to be getting less common compared to jjimjilbangs in Korea. Sentos are bath houses or saunas in urban areas. If you're in the country though, onsens are still popular and these are often outside in natural settings and quite beautiful. These are mostly sex-segregated but there are still a few traditional mixed ones hidden here and there but you're unlikely to stumble across one of those without looking for it.

I struggled with this for a long time when I first started doing long-distance hitchhiking. I get really smelly if I don't do something. I take washcloth showers (which others have explained) but I believe that stopping odor before it starts is crucial!

Clothing material If you allow yourself one luxury, get thin merino wool socks and undies. They're magic. They effectively wick moisture away from your skin, starving odor-causing bacteria. In addition, (unlike most synthetics) merino is intrinsically inhospitable to odor- and rash-causing bacteria. You can wear a pair of undies for days before it smells, but I only do that when I go backcountry hiking. Also, they are unbelievably comfortable! After several ordeals getting foot funguses in Asia I always travel with these.1

Washing Like Wandering Earl, I travel ultralight, so I'm in the habit of scrubbing my shirt armpits, socks, and undies every day. I dry them in my backpack's outer breathable pouch. At midday, I swap them with the socks and undies I've been wearing.

Hair trimming I'm a hairy person, and hair traps odor. I travel with a trimmer and keep things manscaped. It makes a big difference!

(included for completeness) Ammonia-oxidizing bacteria spray Some people don't ever shower and use bacteria sprays that metabolize ammonia from sweat into building blocks which--supposedly-- promote healthy skin. At this time it's prohibitively expensive for me. The NYT has an article about it.

Note, that while @Ankur Banerjee's suggestion to use anti-bacterial soap is effective, be aware that this should not be done regularly! Strong antiseptics annihilate your natural skin flora, which plays a key role in skin health and immunity. Your freshly sterilized skin is then defenceless to potential skin-disease causing bacteria.

1 I am partial to Darn Tough socks (lifetime warranty) and Icebreaker undies. Merino shirts also exist, but they cost a fortune. I bided my time and collected stained or ripped-up shirts at REI garage sales and fixed them up.

Baby wipes. They don't call them "portable showers" for nothing.

Or go for a swim, either in a swimming pool or sea/lake/river.

My solution is typically to use anti-bacterial handwash, the kind that dries up on its own and doesn't need any water at all. When the water supply is cold it's excellent to easily be able to wash up your face and and the bits of your body where you sweat the most.

Generally I'd go for the cold water - a cold shower is a great way to get a bit of a buzz.

If that doesn't appeal, the essential in your pack is wet wipes. With these you can at least sort out the places you most want to feel hygienic, helping to prevent body odour and irritation.