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!
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
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.
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.